Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,23,21,19,22,24);(1,3)(2,5)(4,8)(6,7)(17,19,20,24,21,18,22,23);(10,14)(15,16);(3,8)(5,7)(9,11,13,12)(10,15,14,16)(18,19)(23,24);(2,6)(5,7)(9,12,13,11)(10,16,14,15)(18,19)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=dfi,e^b=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305307, #72305311, #72305319 |
| maximal normal subgroup of | #72305307, #72305311, #72305319 |
| maximal quotient of | #72306186, #72342128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6299061);