Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| 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 | |
742 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)(5,6,10,14,12,7,9,17)(8,19,21,16,24,31,22,13)(11,18,26,27,29,15,25,33)(20,23,34,36,32,30,35,28);(5,7)(6,12)(8,20)(9,17)(10,14)(11,27)(13,30)(15,25)(16,23)(18,26)(19,36)(21,35)(22,34)(24,32)(28,31)(29,33);(2,4)(5,8,10,22,12,24,9,21)(6,13,14,31,7,16,17,19)(11,28,26,30,29,36,25,23)(15,34,33,20,18,35,27,32);(5,9,12,10)(6,14,7,17)(8,21,24,22)(11,25,29,26)(13,31,16,19)(15,33,18,27)(20,35,32,34)(23,36,30,28)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=i,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265587, #72265599, #72265618, … |
| maximal normal subgroup of | #72265587, #72265599, #72265618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406560);