Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,11)(7,10,12,9)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(5,7)(6,10)(8,12)(9,11)(14,18)(17,20)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(7,12)(9,10)(15,19)(17,20)(22,27)(26,33)(28,34)(32,36)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299243);