Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,20,23,22);(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23);(5,6)(7,8)(17,19)(18,22)(20,24)(21,23);(1,3,4,2)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,23)(22,24);(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=i,d^2=fi,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72281019, #72281082, #72281131, … |
| maximal normal subgroup of | #72281019, #72281082, #72281131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 425234);