Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,14,17,13,20)(11,21,25,30,24,32,23,16)(12,26,27,33,28,34,29,18)(22,31,36,39,37,40,38,35);(1,3,4,7)(2,5,6,8)(9,11,15,25,14,24,13,23)(10,16,20,32,17,30,19,21)(12,22,29,38,28,37,27,36)(18,31,26,39,33,40,34,35);(2,6)(5,8)(10,17)(16,30)(18,33)(19,20)(21,32)(26,34)(31,40)(35,39);(9,12,13,27,14,28,15,29)(10,18,19,34,17,33,20,26)(11,22,23,36,24,37,25,38)(16,31,21,35,30,40,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240710, #72240712, #72240714, … |
| maximal normal subgroup of | #72240710, #72240712, #72240714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299605);