Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,26)(16,22)(17,34)(18,24)(19,25)(20,23)(21,29)(27,44)(28,37)(30,43)(31,42)(32,38)(33,41)(35,40)(36,39);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,28)(18,31)(19,29)(20,36)(22,40)(23,37)(24,41)(25,26)(27,33)(32,35)(34,39)(38,43)(42,44);(2,4)(13,16,19,32)(14,22,25,38)(15,27,29,18)(17,33,20,31)(21,24,26,44)(23,42,34,41)(28,35,36,30)(37,40,39,43);(6,10)(9,11)(13,17,19,20)(14,23,25,34)(15,28,29,36)(16,31,32,33)(18,35,27,30)(21,37,26,39)(22,41,38,42)(24,43,44,40);(7,12)(9,11)(13,18,19,27)(14,24,25,44)(15,16,29,32)(17,35,20,30)(21,38,26,22)(23,43,34,40)(28,31,36,33)(37,42,39,41)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574781);