Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,17,18,20)(8,19,13,14)(9,12)(10,15)(11,16);(1,2)(3,4)(6,12)(8,17)(9,10)(13,20)(14,19)(15,16);(2,4)(5,7,9,19,11,18,10,14)(6,13,15,17,16,8,12,20);(5,8,11,13)(6,14,16,19)(7,12,18,15)(9,20,10,17)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285442, #72285620, … |
| maximal normal subgroup of | #72284999, #72285442, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295580);