Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| 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 | |
1438 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)(7,10,14,17)(8,11)(9,12)(13,16)(15,18,20,19);(1,2)(3,4)(5,7,8,15,9,14,16,20)(6,10,13,19,12,17,11,18);(2,4)(10,17)(18,19);(6,11,12,13)(10,18,17,19)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227424, #72252628, #72252977, … |
| maximal normal subgroup of | #72227424, #72252628, #72252977, … |
| maximal quotient of | #72268635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315173);