Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3322 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,20)(16,21)(17,19)(18,23)(22,25)(24,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,16,23)(14,18,21,20)(17,24,25,28)(19,26,22,27);(5,7,8,11)(6,10,9,12)(14,19)(17,25)(18,26)(20,27)(21,22)(24,28);(6,9)(10,12)(13,15)(14,20)(16,23)(17,24)(18,21)(19,27)(22,26)(25,28);(1,3,4,2)(14,21)(15,23)(19,22)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291307);