Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2428 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,22,23,24);(1,2,4,3)(9,10,13,12)(11,15,16,14)(19,23)(22,24);(2,3)(10,12)(14,15)(17,19,21,23)(18,22,20,24);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22);(1,3,4,2)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280903, #72281083, #72281212 |
| maximal normal subgroup of | #72280903, #72281083, #72281212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 903656);