Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14)(15,19)(16,26)(17,29)(18,21)(20,28)(22,31)(23,37)(24,33)(25,41)(27,36)(30,35)(32,38)(34,40)(39,44)(42,43);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,21,33)(16,23,27,39)(17,25,31,40)(20,32,35,43)(22,34,29,41)(26,37,36,44)(28,38,30,42);(1,3,4,2)(6,9)(7,11)(13,16)(14,20)(15,23)(17,30)(18,27)(19,32)(21,35)(22,36)(24,39)(25,42)(26,29)(28,31)(33,43)(34,44)(37,41)(38,40);(5,7,8,11)(6,10,9,12)(15,24)(16,27)(19,33)(20,35)(25,40)(26,36)(28,30)(34,41);(6,9)(10,12)(14,21)(19,33)(20,35)(22,29)(26,36)(32,43)(34,41)(37,44)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421439);