Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,18)(15,20,26,24)(16,21,29,33)(17,22,32,31)(23,34,38,27)(25,35,39,30)(28,36,42,41)(37,43,44,40);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,18,24)(16,27,29,34)(17,30,32,35)(21,23,33,38)(22,25,31,39)(28,37,42,44)(36,43,41,40);(1,2,3,4)(13,16,19,29)(14,21,18,33)(15,23,26,38)(17,28,32,42)(20,34,24,27)(22,36,31,41)(25,37,39,44)(30,40,35,43);(6,10)(9,11)(13,14,19,18)(15,24,26,20)(16,21,29,33)(17,31,32,22)(23,27,38,34)(25,35,39,30)(28,41,42,36)(37,43,44,40);(7,12)(9,11)(13,17,19,32)(14,22,18,31)(15,25,26,39)(16,28,29,42)(20,35,24,30)(21,36,33,41)(23,37,38,44)(27,40,34,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589455);