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 | |
2250 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)(13,14,20,25)(15,21,29,40)(16,22,28,38)(17,32,19,23)(18,24,26,36)(27,39,35,37)(30,41,33,44)(31,43,34,42);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,27)(18,34)(19,35)(20,29)(22,41)(23,37)(24,42)(25,40)(26,31)(28,33)(32,39)(36,43)(38,44);(1,2,3,4)(13,16,20,28)(14,22,25,38)(15,26,29,18)(17,31,19,34)(21,36,40,24)(23,42,32,43)(27,33,35,30)(37,41,39,44);(6,10)(9,11)(13,17,20,19)(14,23,25,32)(15,27,29,35)(16,31,28,34)(18,30,26,33)(21,37,40,39)(22,42,38,43)(24,44,36,41);(7,12)(9,11)(13,18,20,26)(14,24,25,36)(15,28,29,16)(17,33,19,30)(21,38,40,22)(23,41,32,44)(27,31,35,34)(37,42,39,43)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=i,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, 6589919);