Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1707 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,33)(23,29,37,46)(24,38,41,30)(25,31,28,34)(26,32,36,45)(35,51,52,44)(39,50,43,47)(40,49,42,48);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,37)(22,29,33,46)(24,39,41,43)(25,36,28,26)(30,47,38,50)(31,45,34,32)(35,40,52,42)(44,48,51,49);(2,4)(21,24,25,40,27,41,28,42)(22,30,31,48,33,38,34,49)(23,35,26,43,37,52,36,39)(29,44,32,50,46,51,45,47);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,34)(23,36,37,26)(24,40,41,42)(29,45,46,32)(30,48,38,49)(35,39,52,43)(44,47,51,50);(7,15)(10,13)(14,20)(18,19)(21,26,28,23,27,36,25,37)(22,32,34,29,33,45,31,46)(24,39,42,52,41,43,40,35)(30,47,49,51,38,50,48,44)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6716041);