Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C32 | C23 |
| commutator, socle | C22 | C22×C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2)(3,4)(5,6,8,14,9,15,24,39,10,16,25,40,28,43,59,79,11,17,26,41,29,44,60,80,31,46,62,82,65,85,99,114,12,18,27,42,30,45,61,81,32,47,63,83,66,86,100,115,33,48,64,84,67,87,101,116,68,88,102,117,103,118,124,130)(7,13,20,35,21,36,53,73,22,37,54,74,56,76,95,110,23,38,55,75,57,77,96,111,58,78,97,112,98,113,123,129,19,34,49,69,50,70,89,104,51,71,90,105,92,107,119,125,52,72,91,106,93,108,120,126,94,109,121,127,122,128,131,132);(2,4)(5,7)(6,13)(8,20)(9,21)(10,22)(11,23)(12,19)(14,35)(15,36)(16,37)(17,38)(18,34)(24,53)(25,54)(26,55)(27,49)(28,56)(29,57)(30,50)(31,58)(32,51)(33,52)(39,73)(40,74)(41,75)(42,69)(43,76)(44,77)(45,70)(46,78)(47,71)(48,72)(59,95)(60,96)(61,89)(62,97)(63,90)(64,91)(65,98)(66,92)(67,93)(68,94)(79,110)(80,111)(81,104)(82,112)(83,105)(84,106)(85,113)(86,107)(87,108)(88,109)(99,123)(100,119)(101,120)(102,121)(103,122)(114,129)(115,125)(116,126)(117,127)(118,128)(124,131)(130,132);(7,19)(13,34)(20,49)(21,50)(22,51)(23,52)(35,69)(36,70)(37,71)(38,72)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(95,119)(96,120)(97,121)(98,122)(110,125)(111,126)(112,127)(113,128)(123,131)(129,132)Polycyclic presentation:
2^9|a^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^b=ci224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60624);