Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
2030 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)(15,24)(16,22)(17,23)(18,33)(19,34)(20,25)(21,27)(26,38)(28,36)(29,37)(30,43)(31,40)(32,39)(35,41)(42,44);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,27,20,28,17,18)(14,21,23,24,25,33,34,36)(16,26,29,41,31,42,32,30)(22,35,39,38,40,43,37,44);(6,10)(7,11)(13,16)(14,22)(15,26)(17,32)(18,30)(19,29)(20,31)(21,35)(23,39)(24,38)(25,40)(27,41)(28,42)(33,43)(34,37)(36,44);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,34)(15,27,28,18)(16,29,31,32)(21,36,33,24)(22,37,40,39)(26,30,42,41)(35,38,43,44);(13,18,17,28,20,27,19,15)(14,24,34,21,25,36,23,33)(16,30,32,42,31,41,29,26)(22,38,37,35,40,44,39,43)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518945);