Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,18,22)(15,19,26,35)(16,20,28,37)(17,21,30,39)(23,32,25,34)(24,33,41,43)(27,36,42,44)(29,38,31,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,22,35)(16,27,28,42)(17,23,30,25)(20,36,37,44)(21,32,39,34)(24,31,41,29)(33,40,43,38);(1,2,3,4)(15,23)(17,30)(19,32)(21,39)(24,27)(25,26)(29,31)(33,36)(34,35)(38,40)(41,42)(43,44);(5,7,8,11)(6,10,9,12)(16,28)(20,37)(24,41)(27,42)(29,31)(33,43)(36,44)(38,40);(6,9)(10,12)(13,16,18,28)(14,20,22,37)(15,24,26,41)(17,31,30,29)(19,33,35,43)(21,40,39,38)(23,27,25,42)(32,36,34,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6590991);