Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22,28,43,27,32,26,31)(23,33,25,29,36,45,35,46)(24,37,39,49,41,30,40,48)(34,51,42,47,52,50,38,44);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,36)(22,29,32,46)(24,34,41,52)(25,28,35,26)(30,44,37,47)(31,45,43,33)(38,40,42,39)(48,50,49,51);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,34)(25,42)(26,40)(27,41)(28,39)(29,44)(31,49)(32,37)(33,50)(35,38)(36,52)(43,48)(45,51)(46,47);(7,14)(10,13)(15,20)(18,19)(21,25,26,23,27,35,28,36)(22,29,31,33,32,46,43,45)(24,38,40,52,41,42,39,34)(30,47,49,51,37,44,48,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,31,32,43)(23,25,36,35)(24,39,41,40)(29,45,46,33)(30,48,37,49)(34,38,52,42)(44,50,47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=f,e^2=hi,f^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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091174);