Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,28,24,29,23,41)(19,30,33,44,34,43,36,25)(20,35,37,26,32,45,38,46)(21,27,39,42,40,47,31,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,31)(21,38)(22,36)(23,33)(24,34)(26,42)(27,45)(28,43)(29,44)(30,41)(32,39)(35,47)(37,40)(46,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,24,32)(18,26,29,46)(19,31,34,39)(21,36,40,33)(22,37,23,38)(25,42,44,48)(27,30,47,43)(28,45,41,35);(4,12)(7,16)(9,11)(13,15)(17,21,22,39,24,40,23,31)(18,27,28,42,29,47,41,48)(19,32,33,38,34,20,36,37)(25,26,30,45,44,46,43,35);(2,8)(3,10)(7,16)(9,11)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=f,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=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941820);