Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,26)(27,35)(28,33)(29,36);(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)(22,26)(27,35)(28,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293448);