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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,34)(24,30)(25,41)(26,32)(27,33)(28,31)(29,38)(35,52)(36,46)(37,50)(39,47)(40,51)(42,45)(43,49)(44,48);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,29)(24,26)(25,36)(27,38)(28,44)(30,49)(31,46)(32,51)(33,34)(35,37)(39,42)(40,43)(41,48)(45,50)(47,52);(2,4)(21,24,25,39,27,40,28,37)(22,30,41,47,33,51,31,50)(23,35,44,43,38,42,36,26)(29,45,46,32,34,52,48,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,31,33,41)(23,36,38,44)(24,39,40,37)(26,42,43,35)(29,46,34,48)(30,50,51,47)(32,52,49,45);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,25,35,27,43,28,42)(22,32,41,52,33,49,31,45)(23,37,44,24,38,39,36,40)(29,47,46,51,34,50,48,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=hi,d^2=f,e^2=fh,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, 6719118);