Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2188 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)(13,14,19,18)(15,20,25,31)(16,21,27,28)(17,22,30,32)(23,33,37,41)(24,34,38,29)(26,35,40,42)(36,43,44,39);(2,3)(6,10)(11,12)(13,15,19,25)(14,20,18,31)(16,23,27,37)(17,29,30,34)(21,33,28,41)(22,24,32,38)(26,39,40,43)(35,36,42,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,23)(17,26)(18,28)(19,27)(20,33)(22,35)(24,36)(25,37)(29,39)(30,40)(31,41)(32,42)(34,43)(38,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,19,30)(14,22,18,32)(15,24,25,38)(16,26,27,40)(20,34,31,29)(21,35,28,42)(23,36,37,44)(33,43,41,39);(6,10)(11,12)(16,27)(17,30)(21,28)(22,32)(23,37)(24,38)(29,34)(33,41)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=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468573);