Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,10)(9,25)(11,15)(12,16)(13,20)(14,28)(18,21)(19,33)(22,23)(24,27)(26,30)(29,35)(31,36)(32,34);(1,3,4,2)(5,7,11,21,12,22,10,20)(6,13,8,23,16,18,15,17)(9,19,27,35,28,36,26,34)(14,29,24,33,25,32,30,31);(2,3)(5,8,10,6,12,15,11,16)(7,18,20,23,22,13,21,17)(9,24,26,14,28,30,27,25)(19,32,34,33,36,29,35,31);(5,9,10,26,12,28,11,27)(6,14,15,30,16,25,8,24)(7,19,20,34,22,36,21,35)(13,29,17,31,18,32,23,33)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304924);