Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2151 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 136 points:
(5,8)(9,10,16,22)(11,23,29,68)(12,30,35,55)(13,36,41,64)(14,42,43,20)(15,44,46,102)(17,47,52,25)(18,34,56,32)(19,57,60,40)(21,61,63,38)(24,69,73,123)(26,67,28,66)(27,78,79,119)(31,83,89,124)(33,54,92,116)(37,62,98,109)(39,100,101,94)(45,58,77,120)(48,105,107,127)(49,74,51,75)(50,96,110,97)(53,111,114,133)(59,99,95,118)(65,76,122,71)(70,80,104,108)(72,128,129,112)(81,117,113,87)(82,90,103,85)(84,125,132,106)(86,93,88,121)(91,126,130,136)(115,134,135,131);(1,2)(3,5)(4,6)(7,8)(9,11,16,29)(10,17,22,52)(12,24,35,73)(13,37,41,98)(14,26,43,28)(15,45,46,77)(18,48,56,107)(19,58,60,120)(20,49,42,51)(21,62,63,109)(23,64,68,36)(25,57,47,40)(27,61,79,38)(30,80,55,108)(31,84,89,132)(32,71,34,76)(33,91,92,130)(39,96,101,97)(44,50,102,110)(53,112,114,128)(54,115,116,135)(59,119,95,78)(65,121,122,93)(66,100,67,94)(69,124,123,83)(70,117,104,87)(72,90,129,85)(74,118,75,99)(81,126,113,136)(82,125,103,106)(86,131,88,134)(105,133,127,111);(1,3,4,7)(2,5,6,8)(9,12)(10,18)(11,24)(13,31)(14,34)(15,33)(16,35)(17,48)(19,53)(20,55)(21,54)(22,56)(23,65)(25,70)(26,76)(27,72)(28,71)(29,73)(30,42)(32,43)(36,93)(37,84)(38,85)(39,88)(40,87)(41,89)(44,103)(45,91)(46,92)(47,104)(49,108)(50,106)(51,80)(52,107)(57,117)(58,112)(59,113)(60,114)(61,90)(62,115)(63,116)(64,121)(66,123)(67,69)(68,122)(74,105)(75,127)(77,130)(78,126)(79,129)(81,95)(82,102)(83,100)(86,101)(94,124)(96,134)(97,131)(98,132)(99,111)(109,135)(110,125)(118,133)(119,136)(120,128)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=fh,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58488);