Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188035 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2726 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)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72311915, #72311929, #72311963, … |
| maximal normal subgroup of | #72311915, #72311929, #72311963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95239);