Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1911 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)(7,15)(8,17)(9,13)(10,12)(11,14)(16,19)(18,20)(21,22,27,31)(23,32,35,43)(24,29,37,46)(25,30,26,41)(28,40,44,34)(33,48,36,50)(38,47,39,52)(42,51,45,49);(5,7,11,10)(6,12,14,15)(8,18,9,16)(13,20,17,19)(21,23,25,40,27,35,26,34)(22,28,41,43,31,44,30,32)(24,36,38,49,37,33,39,51)(29,45,52,48,46,42,47,50);(1,2,3,4)(5,8,11,9)(6,13,14,17)(7,16,10,18)(12,19,15,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,42)(30,47)(31,46)(32,48)(34,49)(35,36)(40,51)(41,52)(43,50)(44,45);(5,9,11,8)(6,13,14,17)(7,16,10,18)(12,20,15,19)(24,37)(29,46)(33,36)(38,39)(42,45)(47,52)(48,50)(49,51);(5,10,11,7)(6,12,14,15)(8,18,9,16)(13,19,17,20)(21,25,27,26)(22,30,31,41)(23,34,35,40)(24,38,37,39)(28,43,44,32)(29,47,46,52)(33,49,36,51)(42,50,45,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466255);