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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1247 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)(7,11)(8,19)(9,25)(10,15)(12,16)(13,23)(14,28)(17,24)(18,26)(20,22)(21,31)(27,30)(29,34)(32,36)(33,35);(1,3,4,2)(5,7,11,6,12,15,10,16)(8,20,23,17,24,13,22,19)(9,18,27,14,28,30,26,25)(21,33,35,31,36,29,34,32);(2,3)(5,8,11,23,12,24,10,22)(6,13,15,19,16,20,7,17)(9,21,27,35,28,36,26,34)(14,29,30,32,25,33,18,31);(5,9,10,26,12,28,11,27)(6,14,7,18,16,25,15,30)(8,21,22,34,24,36,23,35)(13,29,17,31,20,33,19,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,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, 304925);