Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(2,5)(6,8);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(1,3,4,7)(2,6,5,8);(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213530, C2×#7623096, #72216494, … |
| maximal normal subgroup of | #72213530, C2×#7623096, #72216494, … |
| maximal quotient of | #72213483, C2×#7623096, C4×#33517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530292);