Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,18,23)(15,24,27,20)(16,21,30,39)(17,31,19,22)(25,41,43,34)(26,35,32,36)(28,44,42,37)(29,40,33,38);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,23,24)(16,28,30,42)(17,26,19,32)(21,37,39,44)(22,35,31,36)(25,33,43,29)(34,38,41,40);(1,2,3,4)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,34)(22,38)(23,39)(24,41)(26,42)(27,43)(28,32)(31,40)(35,37)(36,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,31)(15,26,27,32)(16,29,30,33)(20,35,24,36)(21,38,39,40)(25,42,43,28)(34,37,41,44);(6,9)(10,12)(15,26)(17,19)(20,36)(22,31)(24,35)(25,42)(27,32)(28,43)(29,33)(34,44)(37,41)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6590990);