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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2175 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)(7,11)(8,12)(9,10)(13,14,18,19)(15,23,26,20)(16,21,28,29)(17,22,31,32)(24,36,38,33)(25,30,39,34)(27,35,41,42)(37,40,44,43);(2,3)(6,10)(11,12)(13,15,18,26)(14,20,19,23)(16,24,28,38)(17,30,31,34)(21,33,29,36)(22,25,32,39)(27,40,41,43)(35,37,42,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,28)(14,21,19,29)(15,24,26,38)(17,27,31,41)(20,33,23,36)(22,35,32,42)(25,37,39,44)(30,40,34,43);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,25)(16,27)(18,31)(19,32)(20,34)(21,35)(23,30)(24,37)(26,39)(28,41)(29,42)(33,43)(36,40)(38,44);(6,10)(11,12)(16,28)(17,31)(21,29)(22,32)(24,38)(25,39)(30,34)(33,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,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, 7468701);