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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,26)(16,30)(17,33)(18,34)(19,25)(20,23)(21,29)(22,28)(24,36)(27,39)(31,41)(32,42)(35,44)(37,40)(38,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,31)(17,27)(18,32)(19,29)(20,37)(22,41)(23,39)(24,42)(25,26)(28,35)(30,44)(33,40)(34,43)(36,38);(1,2,3,4)(13,16)(14,22)(15,18)(17,32)(19,28)(20,38)(21,34)(23,42)(24,26)(25,30)(27,35)(29,36)(31,37)(33,43)(39,41)(40,44);(6,10)(9,11)(13,17,19,20)(14,23,25,33)(15,27,29,37)(16,32,28,38)(18,35,36,31)(21,39,26,40)(22,42,30,43)(24,44,34,41);(7,12)(9,11)(13,18,19,36)(14,24,25,34)(15,28,29,16)(17,31,20,35)(21,22,26,30)(23,41,33,44)(27,32,37,38)(39,43,40,42)Polycyclic presentation:
2^9|c^2=gi,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, 6589494);