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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1674 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)(21,22,28,25,27,31,26,32)(23,33,42,29,35,48,36,45)(24,30,43,41,39,47,40,38)(34,49,52,44,37,51,50,46);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,29,31,45)(24,37,39,34)(25,33,32,48)(26,42,28,36)(30,46,47,44)(38,49,41,51)(40,50,43,52);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,39)(22,30,31,47)(23,34,35,37)(25,41,32,38)(26,40,28,43)(29,44,45,46)(33,49,48,51)(36,50,42,52);(8,16)(11,20)(13,15)(17,19)(21,25,28,31,27,32,26,22)(23,29,42,48,35,45,36,33)(24,38,43,30,39,41,40,47)(34,46,52,49,37,44,50,51);(6,12)(7,14)(11,20)(13,15)(22,31)(23,35)(24,39)(25,32)(36,42)(40,43)(44,46)(49,51)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,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, 7011977);