Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| 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 | |
892 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)(23,30)(24,34)(25,28)(26,29)(27,33)(31,35)(32,36);(5,6,8,9)(7,10,11,12)(21,23,25,33)(22,27,28,30)(24,32,26,31)(29,36,34,35);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(22,28)(23,31)(24,26)(27,35)(30,36)(32,33);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24,25,26)(22,29,28,34)(23,32,33,31)(27,36,30,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ei,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,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, 287698);