Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,34)(33,36);(6,9)(10,11)(13,15,16,20)(14,17,18,19)(22,26)(23,30)(25,32)(27,35)(29,36)(31,33);(14,18)(17,19)(21,23,24,31)(22,27,26,29)(25,33,32,30)(28,35,34,36);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(22,26)(27,29)(28,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268615);