Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,13,10,14,9,12)(7,15,16,19,17,20,18,11);(5,7,8,18,10,17,9,16)(6,11,13,20,14,19,12,15);(11,19)(15,20);(2,4)(6,12,14,13)(11,15,19,20)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227991, #72268618, #72280214, … |
| maximal normal subgroup of | #72227991, #72268618, #72280214, … |
| maximal quotient of | #72268635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311730);