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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,28)(24,29)(25,30)(26,32)(27,31)(33,41)(34,42)(35,47)(36,43)(37,44)(38,45)(39,46)(40,48)(49,51)(50,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,36)(22,28,31,43)(24,37,39,49)(25,34,26,35)(29,44,46,51)(30,42,32,47)(33,40,50,38)(41,48,52,45);(2,4)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,45)(31,46)(32,48)(34,49)(35,37)(36,50)(42,51)(43,52)(44,47);(6,11)(10,13)(12,19)(18,20)(21,25,27,26)(22,30,31,32)(23,34,36,35)(24,38,39,40)(28,42,43,47)(29,45,46,48)(33,49,50,37)(41,51,52,44);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,31)(23,35)(25,26)(28,42)(29,46)(33,37)(34,36)(38,40)(41,51)(43,47)(44,52)(49,50)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6580842);