Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
938 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)(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,34)(25,30)(26,29)(28,31)(32,36)(33,35);(5,6,8,9)(7,10,11,12)(21,23,31,32,26,24,25,33)(22,27,28,36,29,34,30,35);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,26,23)(22,27,29,34)(25,33,31,32)(28,35,30,36);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,31)(22,28,29,30)(23,32,24,33)(27,35,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=egi,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,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, 288121);