Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
1664 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)(17,18,22,45,24,29,23,28)(19,30,33,25,35,46,34,41)(20,36,38,26,39,40,37,32)(21,27,42,48,44,31,43,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,27)(21,40)(22,34)(23,33)(24,35)(26,42)(28,46)(29,41)(30,45)(31,39)(32,43)(36,44)(37,48)(38,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,38,24,39,23,37)(18,26,28,36,29,32,45,40)(19,31,34,48,35,27,33,47)(21,41,42,30,44,25,43,46);(4,12)(7,16)(9,11)(13,15)(17,21,22,42,24,44,23,43)(18,27,45,48,29,31,28,47)(19,32,33,36,35,26,34,40)(20,25,38,46,39,41,37,30);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,29,45)(19,33,35,34)(20,37,39,38)(21,42,44,43)(25,30,41,46)(26,40,32,36)(27,47,31,48)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=fi,e^2=i,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949290);