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 | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,17)(7,18,23,36)(8,24)(9,26)(10,30,11,16)(13,22,33,21)(14,15)(19,20)(25,34)(27,29)(28,32)(31,35);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,16,18,17,33,30,36)(8,25,26,35,27,32,15,19)(9,28,24,31,14,34,29,20);(2,4)(5,8,11,26,12,27,10,15)(6,14,30,29,17,9,16,24)(7,19,22,25,23,35,21,32)(13,31,36,34,33,20,18,28);(5,9,12,14)(6,15,17,26)(7,20,23,31)(8,16,27,30)(10,29,11,24)(13,32,33,25)(18,35,36,19)(21,34,22,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,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, 295647);