Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| 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 | |
2796 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,27,32)(23,33,35,48)(24,29,39,46)(25,31,26,30)(28,41,43,36)(34,49,37,52)(38,47,40,45)(42,50,44,51);(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,27,35)(22,28,32,43)(24,37,39,34)(25,36,26,41)(29,44,46,42)(30,33,31,48)(38,51,40,50)(45,49,47,52);(2,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,24,27,39)(22,29,32,46)(23,34,35,37)(25,40,26,38)(28,42,43,44)(30,47,31,45)(33,49,48,52)(36,51,41,50);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(23,35)(28,43)(33,48)(34,37)(36,41)(42,44)(49,52)(50,51);(6,14)(11,20)(12,13)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,41)(24,38,39,40)(28,33,43,48)(29,45,46,47)(34,50,37,51)(42,52,44,49)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396764);