Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2825 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,31)(23,33,37,43)(24,29,40,47)(25,32,26,30)(28,35,44,36)(34,49,38,52)(39,48,41,46)(42,51,45,50);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,27,37)(22,28,31,44)(24,38,40,34)(25,36,26,35)(29,45,47,42)(30,33,32,43)(39,50,41,51)(46,49,48,52);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,34)(25,41)(26,39)(27,40)(28,42)(30,48)(31,47)(32,46)(33,49)(35,51)(36,50)(37,38)(43,52)(44,45);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,46,47,48)(34,50,38,51)(42,49,45,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(24,40)(28,44)(30,32)(33,43)(34,38)(39,41)(50,51)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403270);