Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,12,7,11,15)(8,19,22,17,24,20,23,13)(9,25,27,18,28,30,26,14)(21,31,35,33,36,29,34,32);(5,7)(6,12)(8,20)(9,18)(10,15)(11,16)(13,22)(14,28)(17,23)(19,24)(21,33)(25,26)(27,30)(29,35)(31,34)(32,36);(1,2,3,4)(5,8,11,23,12,24,10,22)(6,13,16,19,7,17,15,20)(9,21,26,34,28,36,27,35)(14,29,25,32,18,31,30,33);(5,9,10,27,12,28,11,26)(6,14,16,25,7,18,15,30)(8,21,22,35,24,36,23,34)(13,29,19,32,17,31,20,33)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283983);