Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
765 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)(5,6,9,12,8,11,7,10)(13,14,17,19,18,15,16,20);(5,6,7,10,8,11,9,12)(14,19)(15,20)(16,17);(2,4);(5,7,8,9)(6,10,11,12)(13,15,17,20,18,14,16,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^b=dei,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319843, #72319847, #72320145 |
| maximal normal subgroup of | #72319843, #72319847, #72320145 |
| maximal quotient of | #72321466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136342);