Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,23,27,20)(9,14,29,19)(10,16)(11,17)(12,21)(13,26,35,25)(15,31)(22,32)(24,30,36,28)(33,34);(1,2)(3,4)(5,7,10,21,11,22,31,34)(6,12,15,18,17,33,16,32)(8,24,25,14,27,36,26,19)(9,13,28,23,29,35,30,20);(2,4)(5,8,10,26,11,27,31,25)(6,13,15,20,17,35,16,23)(7,19,21,36,22,14,34,24)(9,18,30,33,29,32,28,12);(5,9)(6,14)(7,20)(8,18)(10,30)(11,29)(12,25)(13,34)(15,24)(16,36)(17,19)(21,35)(22,23)(26,33)(27,32)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295646);