Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2725 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)(21,22,27,39,26,30,25,31)(23,32,38,28,35,45,34,42)(24,29,40,50,36,43,37,44)(33,46,49,41,47,52,48,51);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23,26,35)(22,28,30,42)(24,33,36,47)(25,38,27,34)(29,41,43,51)(31,45,39,32)(37,49,40,48)(44,52,50,46);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,33)(25,37)(26,36)(27,40)(28,41)(30,43)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(42,51)(45,52);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,30)(24,36)(28,42)(31,39)(32,45)(33,47)(37,40)(48,49);(8,16)(11,18)(14,20)(17,19)(24,36)(29,43)(33,47)(37,40)(41,51)(44,50)(46,52)(48,49)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479736);