Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1598 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,12)(7,11,8,10)(17,20);(6,10)(7,8)(11,12)(13,14)(15,17)(16,18)(19,20);(13,15,16,19)(14,17,18,20);(1,3,4,2)(5,7,9,8)(6,11,12,10)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263997, #72264174, #72264184, … |
| maximal normal subgroup of | #72263997, #72264174, #72264184, … |
| maximal quotient of | #72217673, #72217683, C2×#208872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116068);