Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356301 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,27)(19,28,30,42)(20,25,33,39)(21,26,22,36)(24,31,38,32)(29,41,45,48)(34,40,35,46)(37,43,47,44);(17,19,21,31,23,30,22,32)(18,24,36,42,27,38,26,28)(20,29,34,43,33,45,35,44)(25,37,46,48,39,47,40,41);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,43,45,44)(37,41,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263581);