Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 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)(8,16)(11,17)(15,20)(18,19)(21,22,25,31,26,30,42,45)(23,32,35,44,34,28,41,27)(24,36,39,50,40,29,47,49)(33,46,37,43,38,48,51,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,34)(22,27,30,44)(24,37,40,51)(25,41,42,35)(28,45,32,31)(29,48,36,46)(33,47,38,39)(43,50,52,49);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,28)(24,38)(25,35)(26,34)(27,31)(29,48)(30,32)(33,40)(36,46)(37,47)(39,51)(41,42)(43,50)(44,45)(49,52);(7,14)(10,13)(15,20)(18,19)(21,24,25,39,26,40,42,47)(22,29,31,49,30,36,45,50)(23,33,35,37,34,38,41,51)(27,43,32,48,44,52,28,46);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(23,34)(29,36)(31,45)(33,38)(35,41)(37,51)(49,50)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093879);