Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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×#3301 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,23,24,21)(16,22,18,20)(25,28,26,27);(5,6)(7,10)(8,11)(9,12)(14,19)(15,24)(20,22)(25,26);(5,7,9,8)(6,10,12,11)(14,20)(16,18)(19,22)(21,27)(23,28)(25,26);(5,7,9,8)(6,11,12,10)(13,15)(14,21)(16,26)(17,24)(18,25)(19,23)(20,27)(22,28);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,20,19,22)(15,25,24,26)(21,28,23,27)Polycyclic presentation:
2^9|a^2=h,c^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307606);