Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2896 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,27,31)(23,32,35,42)(24,29,37,45)(25,30,26,39)(28,40,43,34)(33,47,50,52)(36,44,38,46)(41,48,51,49);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,27,35)(22,28,31,43)(24,33,37,50)(25,34,26,40)(29,41,45,51)(30,42,39,32)(36,48,38,49)(44,47,46,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,38)(26,36)(27,37)(28,41)(30,46)(31,45)(32,47)(34,49)(35,50)(39,44)(40,48)(42,52)(43,51);(6,11)(10,16)(12,18)(17,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,36,37,38)(28,42,43,32)(29,44,45,46)(33,48,50,49)(41,47,51,52);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(22,31)(28,43)(29,45)(30,39)(32,42)(41,51)(44,46)(47,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343744);