Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20);(1,2,4,3)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20);(10,16)(17,20);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252457, #72252461, D4×#1891, … |
| maximal normal subgroup of | #72252457, #72252461, D4×#1891, … |
| maximal quotient of | #72280639, #72343226 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267355);