Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394177 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,17,19,20)(21,22,27,32)(23,29,34,35)(24,28,33,36)(25,31,26,30);(6,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,24,27,34,26,33)(22,28,30,29,32,36,31,35);(5,7,9,8)(6,11,10,12)(15,19)(17,20)(21,24,25,34,27,33,26,23)(22,29,30,36,32,35,31,28);(2,4)(5,8,9,7)(6,12,10,11)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310251);