Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1430 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)(5,6,10,14,11,15,9,25)(7,16,18,12,20,27,19,28)(8,21,23,29,24,35,22,13)(17,26,31,34,33,36,32,30);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,23,31,24,33,22,32)(13,26,21,34,29,36,35,30);(5,8,9,22,11,24,10,23)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72306933, #72306950, #72306955, … |
| maximal normal subgroup of | #72306933, #72306950, #72306955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300091);