Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #35761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,16)(15,22,26,20)(17,27,19,21)(23,25,28,24);(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,20,16,22)(17,25,19,24)(21,28,27,23);(1,2,3,4)(13,16,18,14)(15,23,26,28)(17,27,19,21)(20,24,22,25);(6,10)(9,11)(13,17,18,19)(14,21,16,27)(15,24,26,25)(20,23,22,28);(7,12)(9,11)(15,25)(17,19)(20,23)(21,27)(22,28)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590061);