Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(8,16)(11,17)(15,20)(18,19)(21,22,26,31,27,32,25,28)(23,29,35,42,33,44,38,40)(24,30,36,45,37,46,39,41)(34,43,49,51,47,52,48,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,33)(24,36)(25,26)(29,42)(31,32)(34,48)(37,39)(40,44)(41,45)(43,52)(47,49);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,27,33)(22,29,32,44)(24,34,37,47)(25,38,26,35)(28,40,31,42)(30,43,46,52)(36,49,39,48)(41,50,45,51);(7,14)(10,13)(15,20)(18,19)(21,24,26,36,27,37,25,39)(22,30,31,45,32,46,28,41)(23,34,35,49,33,47,38,48)(29,43,42,51,44,52,40,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,28)(23,35,33,38)(24,36,37,39)(29,40,44,42)(30,41,46,45)(34,48,47,49)(43,51,52,50)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088477);