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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,32,27,34,26,33)(23,35,25,29,38,47,37,31)(24,39,42,49,44,30,43,50)(36,52,45,48,40,51,41,46);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,38)(22,29,34,31)(24,40,44,36)(25,28,37,26)(30,48,39,46)(32,35,33,47)(41,42,45,43)(49,51,50,52);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,44)(22,30,34,39)(23,36,38,40)(25,45,37,41)(26,43,28,42)(29,46,31,48)(32,50,33,49)(35,51,47,52);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,37,28,38)(22,31,33,47,34,29,32,35)(24,41,43,40,44,45,42,36)(30,46,49,51,39,48,50,52);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,32,34,33)(23,37,38,25)(24,42,44,43)(29,47,31,35)(30,49,39,50)(36,45,40,41)(46,51,48,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=f,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949632);