Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1244 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)(5,6,10,13,9,12,15,7)(8,14,17,20,18,16,19,11);(5,7,10,6,9,13,15,12)(8,16,17,11,18,14,19,20);(5,8)(6,11)(7,14)(9,18)(10,19)(12,20)(13,16)(15,17);(2,4)(5,7,15,12,9,13,10,6)(8,14,17,20,18,16,19,11)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321973 |
| maximal normal subgroup of | #72321973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337508);