Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,14,17,15,11,13,19)(12,16,22,23,20,18,21,24);(5,6,7,8)(9,11,15,10)(12,18,20,16)(13,17,14,19)(21,23,22,24);(1,2,3,4)(9,12,14,22,15,20,13,21)(10,16,17,23,11,18,19,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72209991, #72239937, #72320952, … |
| maximal normal subgroup of | #72209991, #72239937, #72320952, … |
| maximal quotient of | #72209984, #72238285, #72238820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261295);