Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482357 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27,20,24)(21,25,30,32)(22,26)(23,28)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,20)(18,23,26,31)(21,28,30,29)(24,25,27,32);(2,8)(7,15)(9,10)(11,16);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20,22,19)(18,24,26,27)(21,29,30,28)(23,25,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72278264, #72278354, #72278658, … |
| maximal normal subgroup of | #72278264, #72278354, #72278658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389748);