Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 |
| prime radical | #2002 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18)(19,31,34,46)(20,36,33,47)(21,27)(22,29)(23,30)(24,28)(25,35,44,38)(26,39,43,40)(32,42)(37,45)(41,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,34)(18,25,28,44)(20,37,33,32)(21,40,41,39)(22,38,23,35)(26,45,43,42)(27,47,48,36)(29,46,30,31);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,35)(22,39)(23,40)(24,33)(25,42)(27,31)(28,43)(29,36)(30,47)(34,37)(38,41)(44,45)(46,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,37,24,41,23,32)(18,27,29,45,28,48,30,42)(19,33,35,40,34,20,38,39)(25,43,31,47,44,26,46,36);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,28)(19,34)(21,41)(26,43)(29,30)(32,37)(35,38)(36,47)Polycyclic presentation:
2^9|a^2=fg,b^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091198);