Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,10)(8,15)(11,16)(12,17)(14,19)(18,20)(21,22,26,29,25,28,24,34)(23,30,33,35,31,36,32,27);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20)(21,23,24,32,25,31,26,33)(22,27,29,30,28,35,34,36);(1,2,3,4)(5,7,9,13)(6,11,12,18)(8,14,15,19)(10,16,17,20)(22,28)(27,35)(29,34)(30,36);(6,12)(7,13)(10,17)(14,19)(23,31)(27,35)(30,36)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98745);