Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1780 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)(3,7)(4,6)(5,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7);(2,6)(5,7);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72239320, #72239325, #72239327, … |
| maximal normal subgroup of | #72239320, #72239325, #72239327, … |
| maximal quotient of | D4×#2112, C2×#7622691, #72317250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529887);