Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188774 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
592 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)(5,6,10,7,9,13,18,16,11,14,20,17,19,12,8,15);(5,7,10,13,9,16,18,14,11,17,20,12,19,15,8,6);(2,4);(5,8,9,10,11,18,19,20)(6,7,12,15,14,17,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=ei,d^b=dh,e^b=ei,g^d=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239422, #72252782, #72252788 |
| maximal normal subgroup of | #72239422, #72252782, #72252788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402150);