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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 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)(21,22,27,43,26,31,25,32)(23,33,42,28,35,29,36,46)(24,37,44,49,41,30,40,50)(34,47,51,52,39,48,38,45);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,35)(22,28,31,46)(24,38,41,51)(25,42,27,36)(29,43,33,32)(30,48,37,47)(34,44,39,40)(45,50,52,49);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23)(22,29)(24,39)(25,36)(26,35)(27,42)(28,32)(30,48)(31,33)(34,41)(37,47)(38,44)(40,51)(43,46)(45,50)(49,52);(8,16)(11,20)(13,15)(17,19)(21,24,27,44,26,41,25,40)(22,30,43,50,31,37,32,49)(23,34,42,51,35,39,36,38)(28,45,29,47,46,52,33,48);(6,12)(7,14)(11,20)(13,15)(22,31)(23,35)(30,37)(32,43)(34,39)(36,42)(38,51)(49,50)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=hi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011917);