Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2782 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,24)(9,14)(10,16)(11,15)(12,17)(13,21)(19,33)(20,36)(22,28)(23,32)(25,31)(26,30)(27,35)(29,34);(5,7,11,21,12,23,10,22)(6,13,15,32,17,28,16,18)(8,20,9,19,26,34,25,35)(14,29,30,27,31,36,24,33);(1,3,4,2)(5,8,10,25,12,26,11,9)(6,14,15,30,17,31,16,24)(7,19,21,34,23,35,22,20)(13,27,18,29,28,33,32,36);(5,9,11,26,12,25,10,8)(6,14,15,30,17,31,16,24)(7,20,22,35,23,34,21,19)(13,27,18,29,28,33,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=eh,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298326);