Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,15,9,16)(7,12,17,18,13,19,20,8);(2,4)(5,7,10,17,11,13,9,20)(6,12,14,18,15,19,16,8);(5,8,9,19,11,18,10,12)(6,13,14,20,15,7,16,17);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322900 |
| maximal normal subgroup of | #72322900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275226);