Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,17,12,19,28,18)(7,14,23,34,20,32,36,35)(8,24,21,13,26,33,22,31)(9,27,29,15,10,30,25,16);(1,2,3,4)(5,7,10,22,28,36,29,26,12,20,9,21,11,23,25,8)(6,13,15,14,17,33,30,34,19,31,16,32,18,24,27,35);(5,8,25,23,11,21,9,20,12,26,29,36,28,22,10,7)(6,14,30,31,18,35,15,33,19,32,27,13,17,34,16,24);(7,20)(8,26)(13,31)(14,32)(21,22)(23,36)(24,33)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314212, #72314234, #72314256 |
| maximal normal subgroup of | #72314212, #72314234, #72314256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402422);