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 |
| prime radical | #60930 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10)(3,11,13,14)(4,8,7,16)(5,12,15,9)(17,18)(19,28)(20,26)(21,27)(22,24)(23,35)(25,30)(29,43)(31,40)(32,41)(33,38)(34,42)(36,37)(39,46)(44,47)(45,48);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(18,24)(25,37)(26,38)(27,35)(28,40)(39,47)(41,42)(43,48);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19,22,31)(18,25,24,37)(20,29,33,45)(21,30,23,36)(26,39,38,47)(27,40,35,28)(32,44,34,46)(41,43,42,48);(3,13)(4,7)(8,16)(11,14)(17,20)(18,26)(19,29)(21,34)(22,33)(23,32)(24,38)(25,39)(27,42)(28,43)(30,46)(31,45)(35,41)(36,44)(37,47)(40,48);(17,21,22,23)(18,27,24,35)(19,30,31,36)(20,32,33,34)(25,40,37,28)(26,41,38,42)(29,44,45,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,e^2=i,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, 6263590);