Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1261 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)(5,6,11,18,12,16,10,17)(7,19,22,26,23,13,21,33)(8,24,28,34,27,36,32,14)(9,25,29,15,30,20,31,35);(5,7,12,23)(6,13,16,19)(8,25,27,20)(9,14,30,34)(10,22,11,21)(15,28,35,32)(17,33,18,26)(24,29,36,31);(2,4)(5,8,11,32,12,27,10,28)(6,14,17,24,16,34,18,36)(7,20,22,15,23,25,21,35)(9,26,29,19,30,33,31,13);(2,4)(5,9,12,30)(6,15,16,35)(7,14,23,34)(8,26,27,33)(10,31,11,29)(13,32,19,28)(17,25,18,20)(21,36,22,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295565);