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 |
| prime radical | C2×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 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)(5,6,8,9)(7,11,12,10)(13,14,18,23)(15,24,27,20)(16,28,31,21)(17,32,19,22)(25,35,43,41)(26,36,33,37)(29,38,42,44)(30,39,34,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,23,24)(16,29,31,42)(17,26,19,33)(21,38,28,44)(22,36,32,37)(25,34,43,30)(35,39,41,40);(1,2,3,4)(6,9)(10,11)(13,16,18,31)(14,21,23,28)(15,25,27,43)(17,30,19,34)(20,35,24,41)(22,39,32,40)(26,42,33,29)(36,38,37,44);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,22,23,32)(15,26,27,33)(16,30,31,34)(20,36,24,37)(21,39,28,40)(25,42,43,29)(35,38,41,44);(15,26)(17,19)(20,37)(22,32)(24,36)(25,42)(27,33)(29,43)(30,34)(35,44)(38,41)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,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, 6592204);