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 | |
630 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)(5,6,9,15,11,16,10,14)(7,17,19,29,21,28,20,12)(8,22,24,32,26,31,25,13)(18,34,35,30,36,27,23,33);(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,14,28)(8,23,24,18,26,35,25,36)(13,30,22,27,32,33,31,34);(1,3,4,2)(5,8,9,24,11,26,10,25)(6,13,14,31,16,32,15,22)(7,18,20,23,21,36,19,35)(12,27,17,33,29,34,28,30);(7,19,21,20)(12,28,29,17)(18,35,36,23)(27,30,34,33)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=fgh,d^2=ef,e^2=i,f^2=i,g^2=i,h^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, 298253);