Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357620 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,3)(4,6,11,8)(5,9,13,10)(12,14,16,15)(17,18,23,27)(19,28,29,42)(20,25,34,41)(21,26,22,36)(24,31,37,32)(30,43,44,48)(33,39,35,40)(38,46,47,45);(1,3,6,11,7,2,8,4)(5,9,14,12,13,10,15,16)(17,19,22,32,23,29,21,31)(18,24,26,28,27,37,36,42)(20,30,33,45,34,44,35,46)(25,38,40,43,41,47,39,48);(1,4,6,3,7,11,8,2)(5,12,14,10,13,16,15,9)(19,29)(24,37)(28,42)(30,44)(31,32)(38,47)(43,48)(45,46);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(11,16)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,38)(26,40)(27,41)(28,43)(29,44)(31,46)(32,45)(36,39)(37,47)(42,48);(5,13)(9,10)(12,16)(14,15)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,33,34,35)(24,28,37,42)(25,39,41,40)(30,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264853);