Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,18,23)(15,19,25,32)(16,27,28,37)(17,21,30,22)(20,35,36,29)(24,38,40,43)(26,33,41,34)(31,42,44,39);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,31)(21,33)(22,34)(23,32)(27,38)(28,40)(29,39)(30,41)(35,42)(36,44)(37,43);(2,4)(6,9)(10,11)(13,16,17,29,18,28,30,35)(14,20,22,27,23,36,21,37)(15,24,26,39,25,40,41,42)(19,31,34,38,32,44,33,43);(5,7,8,12)(6,10,9,11)(15,25)(16,28)(19,32)(20,36)(26,41)(27,37)(29,35)(33,34);(2,4)(13,17,18,30)(14,21,23,22)(15,26,25,41)(16,29,28,35)(19,33,32,34)(20,37,36,27)(24,39,40,42)(31,43,44,38)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592125);