Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,8,10,13,19,12)(7,14,16,20,18,15,17,11);(5,7)(6,11)(8,15)(9,17)(10,18)(12,14)(13,20)(16,19);(1,2,3,4)(6,12)(8,13)(9,19)(11,15)(14,20)(16,17);(5,8,19,6,10,12,9,13)(7,15,16,11,18,14,17,20)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265528, #72265640, #72265644, … |
| maximal normal subgroup of | #72265528, #72265640, #72265644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290064);