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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2156 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)(17,22,32,33)(24,38,40,34)(25,30,42,35)(27,36,31,37)(39,43,41,44);(2,3)(6,10)(11,12)(13,15,18,26)(14,20,19,23)(16,24,28,40)(17,30,32,35)(21,34,29,38)(22,25,33,42)(27,43,31,44)(36,39,37,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,28)(14,21,19,29)(15,24,26,40)(17,31,32,27)(20,34,23,38)(22,37,33,36)(25,41,42,39)(30,44,35,43);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,32)(14,22,19,33)(15,25,26,42)(16,27,28,31)(20,35,23,30)(21,36,29,37)(24,39,40,41)(34,44,38,43);(6,10)(11,12)(16,28)(17,32)(21,29)(22,33)(24,40)(25,42)(30,35)(34,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,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, 7468955);