Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,4,9)(3,11,13,7,14,16,12,15)(17,18,21,25,22,26,20,30)(19,27,28,23,29,24,31,32);(1,3,6,14)(2,7,8,15)(4,13,5,12)(9,16,10,11)(17,19)(18,23)(20,31)(21,28)(22,29)(24,25)(26,32)(27,30);(17,19)(18,24)(20,28)(21,31)(22,29)(23,25)(26,27)(30,32);(2,8)(7,15)(9,10)(11,16)(17,20,22,21)(18,25,26,30)(19,28,29,31)(23,27,32,24)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306435, #72306442, #72306481, … |
| maximal normal subgroup of | #72306435, #72306442, #72306481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92860);