Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,32)(23,28,24,34)(25,31,26,30)(29,33,36,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,27,24)(22,28,32,34)(25,35,26,33)(29,30,36,31);(21,24,27,23)(22,29,32,36)(25,35,26,33)(28,30,34,31);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(1,3,4,2)(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,33,24,35)(28,29,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,d^b=df,e^b=ei,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307088);