Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,8)(9,10,11,12)(14,18)(15,16)(19,20);(1,2,3,4)(13,14,15,18,17,19,16,20);(5,6,7,8)(14,18)(15,16)(19,20);(14,19)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#23391, #72318166, #72318224 |
| maximal normal subgroup of | C4×#23391, #72318166, #72318224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265660);