Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
914 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)(7,11)(13,14)(15,18)(16,20)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,36)(33,35);(5,6,8,9)(7,12,11,10)(13,15,17,16)(14,18,19,20)(21,23)(22,26)(24,31)(25,30)(27,29)(28,36)(32,33)(34,35);(13,16,17,15)(14,18,19,20)(22,27)(23,30)(25,33)(26,34)(29,36)(31,32);(5,7,8,11)(6,10,9,12)(14,19)(18,20)(22,27)(23,31)(28,35)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=fi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291885);