Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3506 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,21)(14,18)(16,19)(20,24)(22,23);(2,5)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,17)(18,24)(21,23);(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23);(1,3,4,7)(2,6,5,8);(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,16,24,23)Polycyclic presentation:
2^9|d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72317850, #72338532, #72338536, … |
| maximal normal subgroup of | #72317850, #72338532, #72338536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339273);