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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16,10,15,12,28)(7,17,21,32,20,13,22,31)(8,23,25,33,9,26,27,14)(18,30,19,29,36,35,24,34);(1,2,4,3)(5,7,10,20)(6,13,15,17)(8,24,9,19)(11,21,12,22)(14,30,33,35)(16,31,28,32)(18,27,36,25)(23,29,26,34);(5,8,10,9)(6,14,15,33)(7,18,20,36)(11,27,12,25)(13,29,17,34)(16,26,28,23)(19,21,24,22)(30,32,35,31);(1,3,4,2)(5,9,10,8)(6,14,15,33)(7,19,20,24)(11,27,12,25)(13,30,17,35)(16,23,28,26)(18,21,36,22)(29,32,34,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=e,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, 303494);