Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4600 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)(3,5)(4,6)(7,8)(14,20)(21,24);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(3,7)(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(2,6)(5,8);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^9|e^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252477, #72252479, #72252493, … |
| maximal normal subgroup of | #72252477, #72252479, #72252493, … |
| maximal quotient of | #72249909, #72264521, #72302567 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340144);