Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153086 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 132 points:
(5,6,12,18)(7,19,24,65)(8,25,30,51)(9,31,37,62)(10,38,40,16)(11,41,43,102)(13,44,48,21)(14,29,52,27)(15,53,57,36)(17,59,61,34)(20,66,71,124)(22,74,39,64)(23,75,76,121)(26,80,33,96)(28,50,88,116)(32,60,95,122)(35,99,101,90)(42,54,103,119)(45,105,109,127)(46,73,58,94)(47,100,110,93)(49,111,55,120)(56,98,91,117)(63,70,106,68)(67,77,72,107)(69,118,129,112)(78,97,114,84)(79,86,104,82)(81,125,92,108)(83,113,85,89)(87,126,128,130)(115,131,123,132);(1,2)(3,4)(5,7,12,24)(6,13,18,48)(8,20,30,71)(9,32,37,95)(10,39,40,22)(11,42,43,103)(14,45,52,109)(15,54,57,119)(16,58,38,46)(17,60,61,122)(19,62,65,31)(21,53,44,36)(23,59,76,34)(25,77,51,107)(26,81,33,92)(27,70,29,68)(28,87,88,128)(35,100,101,93)(41,47,102,110)(49,112,55,118)(50,115,116,123)(56,121,91,75)(63,113,106,89)(64,90,74,99)(66,96,124,80)(67,97,72,84)(69,86,129,82)(73,98,94,117)(78,126,114,130)(79,125,104,108)(83,132,85,131)(105,120,127,111);(2,4)(5,8)(6,14)(7,20)(9,33)(10,29)(11,28)(12,30)(13,45)(15,55)(16,51)(17,50)(18,52)(19,63)(21,72)(22,70)(23,69)(24,71)(25,38)(26,37)(27,40)(31,89)(32,92)(34,82)(35,83)(36,97)(39,68)(41,104)(42,87)(43,88)(44,67)(46,77)(47,108)(48,109)(49,57)(53,84)(54,118)(56,78)(58,107)(59,86)(60,115)(61,116)(62,113)(64,124)(65,106)(66,74)(73,105)(75,130)(76,129)(79,102)(80,90)(81,95)(85,101)(91,114)(93,131)(94,127)(96,99)(98,120)(100,132)(103,128)(110,125)(111,117)(112,119)(121,126)(122,123)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60292);