Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(8,17)(11,18)(13,19)(16,20);(5,6)(7,13)(8,18)(9,12)(10,16)(11,17)(14,20)(15,19);(2,4)(5,7)(6,10,12,14)(8,16)(9,15)(11,13,18,19)(17,20);(5,6)(7,14)(8,11)(9,12)(10,15)(13,20)(16,19)(17,18)Polycyclic presentation:
2^9|a^2=f,c^2=h,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289764);