Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 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)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,28,34)(23,29,37,46)(24,30,35,44)(25,31,27,33)(26,40,43,32)(36,51,52,45)(38,49,41,47)(39,50,42,48);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,28,37)(22,29,34,46)(24,25,35,27)(26,41,43,38)(30,31,44,33)(32,49,40,47)(36,39,52,42)(45,48,51,50);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,27,23,28,35,25,37)(22,30,33,29,34,44,31,46)(26,38,42,36,43,41,39,52)(32,47,50,45,40,49,48,51);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,28,27)(22,31,34,33)(23,35,37,24)(26,39,43,42)(29,44,46,30)(32,48,40,50)(36,41,52,38)(45,49,51,47);(7,17)(11,14)(15,16)(19,20)(21,26,27,42,28,43,25,39)(22,32,33,50,34,40,31,48)(23,36,35,41,37,52,24,38)(29,45,44,49,46,51,30,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722727);