Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| 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:
(5,6,11,17)(7,18,22,32)(8,24)(9,27)(10,16,12,15)(13,23,33,21)(14,29)(19,35)(20,25)(26,30)(28,34)(31,36);(1,2)(3,4)(5,7,10,21,11,22,12,23)(6,13,15,18,17,33,16,32)(8,25,14,19,27,36,26,34)(9,28,29,31,24,35,30,20);(2,4)(5,8,12,14,11,27,10,26)(6,9,16,29,17,24,15,30)(7,19,23,36,22,34,21,25)(13,31,32,35,33,20,18,28);(5,9,11,24)(6,14,17,26)(7,20,22,31)(8,16,27,15)(10,30,12,29)(13,25,33,36)(18,34,32,19)(21,35,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 295657);