Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2922 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)(23,32,35,41)(24,29,26,39)(25,30,38,46)(28,40,43,33)(34,47,50,51)(36,44,37,45)(42,49,52,48);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,49)(38,50)(44,51)(45,47)(46,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,27,26)(22,29,31,39)(23,33,35,40)(25,37,38,36)(28,41,43,32)(30,45,46,44)(34,49,50,48)(42,47,52,51);(6,11)(10,16)(12,18)(17,20)(21,25,27,38)(22,30,31,46)(23,34,35,50)(24,36,26,37)(28,42,43,52)(29,44,39,45)(32,47,41,51)(33,48,40,49);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(25,38)(30,46)(34,50)(36,37)(42,52)(44,45)(47,51)(48,49)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343786);