Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,20,26,23)(16,17,21,18)(24,25,28,27);(6,9)(7,11)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,28)(22,23);(2,4)(13,16,19,21)(14,17,22,18)(15,23,26,20)(24,25,28,27);(6,9)(10,12)(13,16)(14,17)(15,24)(18,22)(19,21)(20,25)(23,27)(26,28);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(16,22)(18,19)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6580236);