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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(21,22,27,33,28,34,26,32)(23,35,37,31,38,47,25,29)(24,30,40,45,41,46,42,48)(36,49,50,44,51,52,39,43);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,38)(22,29,34,31)(24,36,41,51)(25,26,37,27)(30,43,46,44)(32,35,33,47)(39,42,50,40)(45,52,48,49);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,30)(23,36)(25,39)(26,42)(27,40)(28,41)(29,43)(31,44)(32,48)(33,45)(34,46)(35,49)(37,50)(38,51)(47,52);(8,16)(11,20)(13,15)(17,19)(21,25,27,23,28,37,26,38)(22,31,33,47,34,29,32,35)(24,39,40,36,41,50,42,51)(30,44,45,52,46,43,48,49);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,32,34,33)(23,37,38,25)(24,40,41,42)(29,35,31,47)(30,45,46,48)(36,39,51,50)(43,52,44,49)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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949664);