Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201938 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9)(3,5,8,15)(4,11,14,10)(7,13,16,12)(17,18,21,24,22,26,20,25)(19,27,28,23,29,31,30,32);(1,3,5,9,6,8,15,2)(4,10,13,16,14,11,12,7)(17,19)(18,23,26,32)(20,30)(21,28)(22,29)(24,31,25,27);(1,4)(2,7)(3,10)(5,13)(6,14)(8,11)(9,16)(12,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,30);(1,5,6,15)(2,8,9,3)(4,12,14,13)(7,10,16,11)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286423, #72286729, #72286796, … |
| maximal normal subgroup of | #72286423, #72286729, #72286796, … |
| maximal quotient of | C2×#202089, #72309101, #72309120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109285);