Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #95218 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,14,18,13,16)(11,19,21,15,22,23,20,24)(25,26)(27,33,35,30)(28,36)(29,32)(31,38)(34,39,37,40);(1,3)(2,5,6,7)(4,8)(9,11,13,20,14,22,12,21)(10,15,17,23,18,24,16,19)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(2,6)(5,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=gi,f^2=gi,b^a=bd,c^a=cgh,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239123 |
| maximal normal subgroup of | #72239123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2541);