Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,17,12,18)(7,15,9,26,20,29,21,13)(8,22,24,32,25,14,27,31)(19,33,23,28,34,30,35,36);(5,7)(6,13)(8,23)(9,12)(10,21)(11,20)(14,30)(15,18)(16,29)(17,26)(19,24)(22,33)(25,35)(27,34)(28,31)(32,36);(1,2,3,4)(5,8,11,25)(6,14,17,22)(7,19,20,34)(9,23,21,35)(10,24,12,27)(13,28,26,36)(15,30,29,33)(16,31,18,32);(5,9,12,7,11,21,10,20)(6,15,18,13,17,29,16,26)(8,23,27,19,25,35,24,34)(14,30,32,28,22,33,31,36)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331028);