Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1696 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,44,27,33,26,32)(23,34,36,29,24,30,37,46)(25,39,41,50,43,31,42,49)(35,47,51,52,38,48,40,45);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,24)(22,29,33,46)(25,40,43,51)(26,36,28,37)(30,44,34,32)(31,48,39,47)(35,41,38,42)(45,49,52,50);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,23)(22,30,33,34)(25,38,43,35)(26,36,28,37)(29,44,46,32)(31,47,39,48)(40,42,51,41)(45,49,52,50);(8,16)(11,20)(13,15)(17,19)(21,25,28,41,27,43,26,42)(22,31,44,49,33,39,32,50)(23,35,36,51,24,38,37,40)(29,45,30,47,46,52,34,48);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,32,33,44)(23,36,24,37)(25,41,43,42)(29,30,46,34)(31,49,39,50)(35,40,38,51)(45,48,52,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=hi,d^2=fi,e^2=i,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011998);