Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,27)(23,26,31,34)(24,28,32,36)(29,33,30,35);(6,9)(10,12)(21,23,24,30,25,31,32,29)(22,26,28,35,27,34,36,33);(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,19,20)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(1,3,4,2)(7,11)(10,12)(13,16,17,15)(14,18,19,20)(22,27)(26,34)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289498);