Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| 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 | #2358 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1567 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)(5,6,10,15,11,16,9,12)(7,17,20,13,21,27,19,28)(8,14,22,30,24,29,23,25)(18,31,32,26,33,36,34,35);(1,2)(3,4)(6,12,16,15)(8,22,24,23)(13,17,28,27)(14,29)(18,32,33,34)(25,30)(26,35)(31,36);(5,7,10,20,11,21,9,19)(6,13,15,27,16,28,12,17)(8,18,22,32,24,33,23,34)(14,26,30,36,29,35,25,31);(2,4)(5,8,10,22,11,24,9,23)(6,14,15,30,16,29,12,25)(7,18,20,32,21,33,19,34)(13,26,27,36,28,35,17,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,f^c=fi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280601, #72280718 |
| maximal normal subgroup of | #72280601, #72280718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263876);