Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,9,13,10,14,8,12)(7,15,18,19,17,20,16,11);(1,2,4,3)(5,7,8,16,10,17,9,18)(6,11,13,15,14,19,12,20);(11,19)(15,20);(1,3,4,2)(6,12,14,13)(11,20,19,15)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250083, #72252980, #72256917, … |
| maximal normal subgroup of | #72250083, #72252980, #72256917, … |
| maximal quotient of | #72308364, #72343215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283335);