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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(21,22,27,32)(23,28,35,43)(24,29,38,46)(25,30,26,31)(33,41,50,52)(34,42,39,47)(36,44,49,51)(37,45,40,48);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,28)(24,36)(25,34)(26,39)(27,35)(29,44)(30,42)(31,47)(32,43)(33,37)(38,49)(40,50)(41,45)(46,51)(48,52);(2,4)(21,24)(22,29)(23,33)(25,37)(26,40)(27,38)(28,41)(30,45)(31,48)(32,46)(34,49)(35,50)(36,39)(42,51)(43,52)(44,47);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,26)(22,30,32,31)(23,34,35,39)(24,37,38,40)(28,42,43,47)(29,45,46,48)(33,49,50,36)(41,51,52,44);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(25,26)(28,42)(30,31)(33,49)(35,39)(36,50)(37,40)(41,51)(43,47)(44,52)(45,48)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6584738);