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 | |
2238 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)(13,14,17,21)(15,23,27,19)(16,20,28,34)(18,22,30,36)(24,33,26,31)(25,37,39,32)(29,35,41,44)(38,43,40,42);(5,7)(6,9)(8,12)(10,11)(13,15,17,27)(14,19,21,23)(16,25,28,39)(18,24,30,26)(20,32,34,37)(22,31,36,33)(29,38,41,40)(35,42,44,43);(1,2,3,4)(15,24)(18,30)(19,31)(22,36)(23,33)(25,38)(26,27)(29,41)(32,42)(35,44)(37,43)(39,40);(6,10)(9,11)(13,16)(14,20)(15,25)(17,28)(18,29)(19,32)(21,34)(22,35)(23,37)(24,38)(26,40)(27,39)(30,41)(31,42)(33,43)(36,44);(7,12)(9,11)(15,26)(16,28)(18,30)(19,33)(20,34)(22,36)(23,31)(24,27)(25,38)(32,42)(37,43)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6589920);