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 | C2×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,7,14,15,13,16)(17,18,22,26,23,27,21,31)(19,25,28,24,20,29,30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19,23,20)(18,24,27,32)(21,30,22,28)(25,26,29,31);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,16,11)(17,20,23,19)(18,25,27,29)(21,30,22,28)(24,26,32,31);(2,9)(7,16)(8,10)(11,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,25,32,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,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 | #72306403, #72306428, #72306442, … |
| maximal normal subgroup of | #72306403, #72306428, #72306442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92863);