Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1700 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)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,31)(23,32,35,28)(24,29,38,47)(25,39,27,30)(33,49,51,42)(34,43,40,44)(36,52,50,45)(37,48,41,46);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,35)(22,28,31,32)(24,36,38,50)(25,34,27,40)(29,45,47,52)(30,43,39,44)(33,41,51,37)(42,46,49,48);(1,2,3,4)(21,24)(22,29)(23,33)(25,37)(26,38)(27,41)(28,42)(30,46)(31,47)(32,49)(34,50)(35,51)(36,40)(39,48)(43,45)(44,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,27)(22,30,31,39)(23,34,35,40)(24,37,38,41)(28,43,32,44)(29,46,47,48)(33,50,51,36)(42,45,49,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(25,27)(28,44)(30,39)(32,43)(33,50)(35,40)(36,51)(37,41)(42,52)(45,49)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593819);