Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,11,10,15,24,14)(7,16,19,28,20,25,18,12)(8,13,21,26,23,29,22,30)(17,31,33,36,34,35,32,27);(6,11,15,14)(8,21,23,22)(12,25,28,16)(17,32,34,33);(1,2,4,3)(5,7,10,20)(6,12,15,28)(8,17,23,34)(9,18,24,19)(11,25,14,16)(13,27,29,36)(21,32,22,33)(26,35,30,31);(1,3,4,2)(5,8,10,23)(6,13,15,29)(7,17,20,34)(9,21,24,22)(11,26,14,30)(12,27,28,36)(16,31,25,35)(18,32,19,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,f^c=fi,g^c=gi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224693, #72312207, #72312252, … |
| maximal normal subgroup of | #72224693, #72312207, #72312252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264194);