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 |
| prime radical | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,27,31,21)(16,33,34,43)(17,36,37,39)(18,26,20,24)(22,35,42,29)(23,28,44,38)(30,40,32,41);(2,4)(6,9)(10,12)(13,15)(14,21)(16,17)(18,30)(19,31)(20,32)(22,23)(24,40)(25,27)(26,41)(28,29)(33,39)(34,37)(35,38)(36,43)(42,44);(5,7,8,11)(6,10,9,12)(13,16,20,35,19,34,18,29)(14,22,26,43,25,42,24,33)(15,28,30,37,31,38,32,17)(21,39,40,44,27,36,41,23);(1,2,3,4)(13,17,20,28,19,37,18,38)(14,23,26,39,25,44,24,36)(15,29,30,16,31,35,32,34)(21,33,40,22,27,43,41,42);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,30,31,32)(16,29,34,35)(17,28,37,38)(21,40,27,41)(22,33,42,43)(23,39,44,36)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7520914);