Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1107 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)(19,25)(20,28,29,23)(21,24)(22,26,30,32)(27,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,21,27)(18,22,24,30)(20,26,29,32)(23,25,28,31);(2,8)(7,15)(9,10)(11,16);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281887, #72303901, #72304004, … |
| maximal normal subgroup of | #72281887, #72303901, #72304004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389147);