Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
2716 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)(13,14,19,25)(15,26,30,21)(16,31,29,22)(17,34,20,23)(18,36,27,24)(28,39,37,40)(32,41,35,44)(33,42,38,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,30)(14,21,25,26)(16,32,29,35)(17,28,20,37)(18,33,27,38)(22,41,31,44)(23,39,34,40)(24,42,36,43);(2,4)(13,16,19,29)(14,22,25,31)(15,27,30,18)(17,33,20,38)(21,24,26,36)(23,42,34,43)(28,35,37,32)(39,41,40,44);(6,10)(9,11)(13,17,19,20)(14,23,25,34)(15,28,30,37)(16,33,29,38)(18,32,27,35)(21,39,26,40)(22,42,31,43)(24,41,36,44);(7,12)(9,11)(13,18,19,27)(14,24,25,36)(15,29,30,16)(17,35,20,32)(21,22,26,31)(23,44,34,41)(28,33,37,38)(39,43,40,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,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,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575031);