Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
2021 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)(7,9)(8,10)(11,12)(13,14,20,25)(15,26,18,33)(16,22,31,39)(17,23,19,35)(21,28,24,29)(27,41,34,44)(30,37,32,38)(36,43,40,42);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,28,20,18,19,29)(14,21,35,33,25,24,23,26)(16,27,32,43,31,34,30,42)(22,36,37,44,39,40,38,41);(6,10)(7,11)(13,16)(14,22)(15,27)(17,32)(18,34)(19,30)(20,31)(21,36)(23,38)(24,40)(25,39)(26,41)(28,43)(29,42)(33,44)(35,37);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,35)(15,28,18,29)(16,30,31,32)(21,26,24,33)(22,37,39,38)(27,42,34,43)(36,44,40,41);(13,18,19,28,20,15,17,29)(14,24,23,33,25,21,35,26)(16,27,30,42,31,34,32,43)(22,36,38,41,39,40,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=hi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518912);