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 | |
1687 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)(17,18,21,25,22,27,29,26)(19,28,32,23,31,38,42,37)(20,24,34,33,35,40,43,39)(30,41,46,36,45,44,48,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18)(19,23)(20,33)(21,26)(22,27)(24,34)(25,29)(28,32)(30,44)(31,37)(35,39)(36,46)(38,42)(40,43)(41,45)(47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19,29,42,22,31,21,32)(18,23,25,38,27,37,26,28)(20,30,43,48,35,45,34,46)(24,36,33,44,40,47,39,41);(4,12)(7,16)(9,11)(13,15)(17,20,29,43,22,35,21,34)(18,24,26,39,27,40,25,33)(19,30,42,48,31,45,32,46)(23,36,28,41,37,47,38,44);(2,8)(3,10)(7,16)(9,11)(19,31)(23,37)(28,38)(30,45)(32,42)(36,47)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=fh,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=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949293);