Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #25553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 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)(21,22,25,30,26,31,39,27)(23,32,35,40,34,28,42,44)(24,29,36,46,38,47,50,41)(33,45,48,51,37,43,49,52);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(24,36)(25,39)(28,40)(30,31)(32,44)(33,48)(35,42)(37,49)(38,50)(41,46)(51,52);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,26,34)(22,28,31,32)(24,37,38,33)(25,35,39,42)(27,40,30,44)(29,45,47,43)(36,49,50,48)(41,52,46,51);(8,16)(11,20)(13,15)(17,19)(21,24,25,36,26,38,39,50)(22,29,30,46,31,47,27,41)(23,33,35,48,34,37,42,49)(28,43,44,52,32,45,40,51);(6,12)(7,14)(11,20)(13,15)(23,34)(28,32)(33,37)(35,42)(40,44)(43,45)(48,49)(51,52)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011576);