Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1471 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)(5,6,10,16,11,12,9,15)(7,17,19,8,18,20,14,13);(6,12)(7,18)(9,10)(17,20);(2,4)(5,7,10,14,11,18,9,19)(6,13,15,17,12,8,16,20);(2,4)(5,8)(6,14)(7,16)(9,17)(10,20)(11,13)(12,19)(15,18)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338561, #72338563, #72338566, … |
| maximal normal subgroup of | #72338561, #72338563, #72338566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295562);