Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,26,18,27,9,14,11,16,30,15,12,8,24,29)(7,17,28,32,31,36,21,19,20,34,22,13,23,33,35,25);(1,2)(3,4)(5,7,18,23,11,20,12,31)(6,13,8,25,16,32,27,19)(9,28,30,35,24,22,10,21)(14,33,26,17,29,36,15,34);(2,4)(5,8)(6,12)(7,19)(9,29)(10,26)(11,27)(13,23)(14,24)(15,30)(16,18)(17,35)(20,25)(21,34)(22,36)(28,33)(31,32);(7,20)(13,32)(17,34)(19,25)(21,35)(22,28)(23,31)(33,36)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72224939, #72226807, … |
| maximal normal subgroup of | #72224928, #72224939, #72226807, … |
| maximal quotient of | #72224921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406737);