Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,14,10,11,8,13)(7,15,17,19,16,20,18,12);(1,2)(3,4)(6,11)(7,16)(8,9)(12,19);(2,4)(5,7,8,17,10,16,9,18)(6,12,14,20,11,19,13,15);(6,13,11,14)(12,15,19,20)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286681, #72287267, #72287360, … |
| maximal normal subgroup of | #72286681, #72287267, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291622);