Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
2137 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)(7,10)(8,9)(11,12)(13,14)(15,26)(16,29)(17,32)(18,34)(19,25)(20,23)(21,28)(22,31)(24,36)(27,38)(30,39)(33,40)(35,41)(37,44)(42,43);(6,9)(7,11)(13,15)(14,21)(16,17)(18,35)(19,28)(20,31)(22,32)(23,29)(24,41)(25,26)(27,33)(30,36)(34,39)(37,43)(38,44)(40,42);(1,2,3,4)(13,16,17,15,19,31,20,28)(14,22,32,21,25,29,23,26)(18,30,33,27,36,35,37,43)(24,39,44,38,34,41,40,42);(6,9)(10,12)(13,17,19,20)(14,23,25,32)(15,16,28,31)(18,33,36,37)(21,29,26,22)(24,40,34,44)(27,30,43,35)(38,41,42,39);(5,7,8,11)(6,10,9,12)(13,18,17,33,19,36,20,37)(14,24,32,44,25,34,23,40)(15,27,31,35,28,43,16,30)(21,38,29,41,26,42,22,39)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723068);