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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,26,31,27,32,25,28)(23,33,36,42,34,29,40,44)(24,30,37,47,39,48,41,43)(35,46,50,51,38,45,49,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,34)(24,37)(25,26)(29,44)(31,32)(33,42)(35,49)(38,50)(39,41)(43,47)(45,46);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,27,34)(22,29,32,33)(24,38,39,35)(25,40,26,36)(28,42,31,44)(30,46,48,45)(37,49,41,50)(43,52,47,51);(7,14)(10,13)(15,20)(18,19)(21,24,26,37,27,39,25,41)(22,30,31,47,32,48,28,43)(23,35,36,50,34,38,40,49)(29,45,44,52,33,46,42,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,28)(23,36,34,40)(24,37,39,41)(29,42,33,44)(30,43,48,47)(35,49,38,50)(45,52,46,51)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,d^2=f,e^2=h,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=dh,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, 7094031);