Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,30,16,20,14,19)(11,21,29,17,24,33,23,34)(12,25,13,18,27,38,26,35)(22,31,39,32,37,40,28,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,22)(13,28)(14,29)(15,23)(16,24)(18,31)(19,33)(20,34)(21,30)(25,32)(26,39)(27,37)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12,14,26,16,27,15,13)(10,18,30,38,20,35,19,25)(11,22,29,39,24,37,23,28)(17,31,21,36,34,40,33,32);(3,8)(5,7)(9,13,15,27,16,26,14,12)(10,18,30,38,20,35,19,25)(11,22,29,39,24,37,23,28)(17,32,33,40,34,36,21,31);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928552);