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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,16,17,14)(7,13,19,11,20,28,26,29)(8,21,23,12,25,32,24,33)(18,27,22,31,35,36,34,30);(1,2)(3,4)(5,7,17,26,10,20,9,19)(6,11,15,28,16,29,14,13)(8,22,24,18,25,34,23,35)(12,30,32,27,33,31,21,36);(2,4)(5,8,10,25)(6,12)(7,18,20,35)(9,24,17,23)(11,27)(13,31)(14,32)(15,21)(16,33)(19,34,26,22)(28,30)(29,36);(5,7,10,20)(6,13,16,28)(8,18,25,35)(9,26,17,19)(11,15,29,14)(12,31,33,30)(21,36,32,27)(22,23,34,24)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=eh,d^2=i,e^2=i,f^2=i,g^2=i,h^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, 295651);