Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35821 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2066 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)(13,14,17,18)(15,22,25,19)(16,20,26,21)(23,24,28,27);(5,7,8,12)(6,9,10,11)(13,15,17,25)(14,19,18,22)(16,24,26,27)(20,23,21,28);(1,2,3,4)(13,16)(14,20)(15,23)(17,26)(18,21)(19,27)(22,24)(25,28);(6,10)(9,11)(13,16)(14,21)(15,24)(17,26)(18,20)(19,28)(22,23)(25,27);(7,12)(9,11)(13,14)(15,25)(16,20)(17,18)(21,26)(23,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6590093);