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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 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)(16,28)(17,32)(18,22)(19,23)(21,30)(24,38)(25,36)(26,35)(27,37)(29,34)(31,39)(33,40)(41,43)(42,44);(2,3)(6,9)(10,12)(13,15,18,27)(14,20,22,37)(16,29,30,24)(17,26,19,25)(21,38,28,34)(23,36,32,35)(31,41,33,42)(39,43,40,44);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,33)(18,30)(19,31)(20,34)(22,28)(23,40)(25,42)(26,41)(27,29)(32,39)(35,44)(36,43)(37,38);(1,3,4,2)(14,22)(15,25)(16,30)(17,19)(20,35)(24,41)(26,27)(29,42)(34,43)(36,37)(38,44)(39,40);(6,9)(10,12)(13,17,18,19)(14,23,22,32)(15,26,27,25)(16,31,30,33)(20,36,37,35)(21,39,28,40)(24,42,29,41)(34,44,38,43)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=gi,e^2=i,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, 7467858);