Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,21,20,24);(10,11)(14,15)(18,23)(19,20);(1,2,4,3)(5,6)(7,8)(17,19,22,20)(18,24,23,21);(1,3,4,2)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,21,23,24);(9,12)(10,14)(11,15)(13,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=ghi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232057, #72232060, #72232061, … |
| maximal normal subgroup of | #72232057, #72232060, #72232061, … |
| maximal quotient of | #72323539, C2×#524651, #72331264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 431847);