Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,12)(11,14,15,16)(17,18,22,21)(19,20,24,23);(5,6,7,8)(10,12)(14,16)(17,19,22,24)(18,23,21,20);(1,2)(3,4)(5,6)(7,8);(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23);(9,12,13,10)(11,14,15,16)(17,21,22,18)(19,23,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72297354, #72297363, #72297399, … |
| maximal normal subgroup of | #72297354, #72297363, #72297399, … |
| maximal quotient of | #72298197, #72298207, #72298221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 697331);