Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 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)(8,16)(11,17)(15,20)(18,19)(21,22)(23,34)(24,39)(25,31)(26,32)(27,44)(28,33)(29,37)(30,41)(35,51)(36,48)(38,52)(40,49)(42,46)(43,47)(45,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,36,28,38,27,37)(22,29,44,52,33,48,32,34)(24,35,43,42,41,25,40,45)(30,46,47,51,39,50,49,31);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,38)(26,43)(27,40)(28,41)(29,46)(31,34)(32,49)(33,39)(36,42)(37,45)(44,47)(48,50)(51,52);(7,14)(10,13)(15,20)(18,19)(21,25,28,35)(22,31,33,51)(23,24,38,41)(26,42,27,45)(29,47,48,49)(30,34,39,52)(32,46,44,50)(36,40,37,43);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,33,44)(23,36,38,37)(24,40,41,43)(25,42,35,45)(29,34,48,52)(30,47,39,49)(31,46,51,50)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089904);