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 | |
2744 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)(16,30)(17,35)(18,37)(19,25)(20,23)(21,29)(22,33)(24,27)(28,39)(31,41)(32,42)(34,43)(36,44)(38,40);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,31)(17,28)(18,32)(19,29)(20,38)(22,41)(23,39)(24,42)(25,26)(27,34)(30,44)(33,36)(35,40)(37,43);(2,4)(13,16,19,33)(14,22,25,30)(15,27,29,18)(17,34,20,32)(21,24,26,37)(23,43,35,42)(28,36,38,31)(39,41,40,44);(6,10)(9,11)(13,17,19,20)(14,23,25,35)(15,28,29,38)(16,32,33,34)(18,36,27,31)(21,39,26,40)(22,42,30,43)(24,44,37,41);(7,12)(9,11)(13,18,19,27)(14,24,25,37)(15,16,29,33)(17,36,20,31)(21,30,26,22)(23,44,35,41)(28,32,38,34)(39,43,40,42)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,e^a=ei,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, 6574829);