Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
2039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,25,17,23)(15,20,29,37,30,38,28,36)(16,22,31,39,32,41,33,21)(26,40,42,44,27,35,43,34);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,28)(18,29)(19,30)(21,34)(22,40)(23,36)(24,37)(25,38)(27,32)(31,42)(33,43)(35,41)(39,44);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,23)(24,41)(25,39)(27,30)(28,42)(29,43)(35,37)(36,40)(38,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,27)(17,33)(18,31)(19,32)(20,35)(21,23)(24,39)(25,41)(26,30)(28,42)(29,43)(34,37)(36,44)(38,40);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,31,32,33)(20,36,38,37)(21,22,39,41)(26,42,27,43)(34,40,44,35)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913524);