Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ 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:
(1,2)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,31)(23,32,35,43)(24,29,38,47)(25,30,26,41)(28,39,44,34)(33,49,36,52)(37,46,40,48)(42,50,45,51);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,34,27,35,25,39)(22,28,30,32,31,44,41,43)(24,36,37,50,38,33,40,51)(29,45,48,52,47,42,46,49);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,33)(25,40)(26,37)(27,38)(28,42)(30,48)(31,47)(32,49)(34,51)(35,36)(39,50)(41,46)(43,52)(44,45);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,37,38,40)(28,43,44,32)(29,46,47,48)(33,50,36,51)(42,49,45,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(28,44)(29,47)(30,41)(32,43)(42,45)(46,48)(49,52)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403650);