Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1842 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)(3,4)(5,6,12,16,10,9,11,15)(7,17,21,29,19,13,20,30)(8,22,27,14,25,32,26,24)(18,31,23,33,34,36,35,28);(1,3,4,2)(5,7,10,19)(6,13,9,17)(8,23,25,35)(11,20,12,21)(14,31,24,36)(15,29,16,30)(18,27,34,26)(22,28,32,33);(2,3)(5,8)(6,14)(7,18)(9,24)(10,25)(11,27)(12,26)(13,28)(15,32)(16,22)(17,33)(19,34)(20,23)(21,35)(29,31)(30,36);(5,9,10,6)(7,17,19,13)(8,24,25,14)(11,16,12,15)(18,33,34,28)(20,30,21,29)(22,26,32,27)(23,36,35,31)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 302924);