Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,34)(26,35)(27,37)(28,36)(29,38)(30,39)(31,40)(32,42)(33,41)(43,44);(2,3)(6,9)(10,12)(13,15,19,28)(14,20,23,36)(16,25,17,30)(18,32,33,27)(21,34,22,39)(24,42,41,37)(26,29,43,31)(35,38,44,40);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,19)(18,29)(20,34)(22,23)(24,38)(26,27)(28,30)(31,33)(32,43)(35,37)(36,39)(40,41)(42,44);(1,3,4,2)(13,17,19,16)(14,22,23,21)(15,26,28,43)(18,29,33,31)(20,35,36,44)(24,38,41,40)(25,32,30,27)(34,42,39,37);(6,9)(10,12)(14,23)(16,17)(20,36)(24,41)(25,30)(26,43)(29,31)(37,42)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467853);