Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,8)(9,10,12,11)(13,14,17,20)(15,18,16,19);(2,4)(9,11,12,10)(14,18)(15,16)(19,20);(5,6,7,8);(14,19)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^b=gi,e^d=ei,g^d=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305525, D4×#1864, #72305588, … |
| maximal normal subgroup of | #72305525, D4×#1864, #72305588, … |
| maximal quotient of | #72309799 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300128);