Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,12,17)(7,18,22,9)(8,23,27,31)(10,16,11,15)(13,21,28,20)(14,29,32,26)(19,34,35,30)(24,33,36,25);(5,7,11,21,12,22,10,20)(6,13,16,9,17,28,15,18)(8,24,26,35,27,36,29,19)(14,25,23,30,32,33,31,34);(5,8,11,29,12,27,10,26)(6,14,15,23,17,32,16,31)(7,19,21,36,22,35,20,24)(9,25,13,30,18,33,28,34);(5,9,11,13,12,18,10,28)(6,7,15,21,17,22,16,20)(8,25,29,30,27,33,26,34)(14,19,23,36,32,35,31,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=efg,d^2=efg,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,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295313);