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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22,27,31)(23,28,26,41)(24,34,36,50)(25,30,39,46)(29,32,44,37)(33,42,40,52)(35,49,38,51)(43,47,45,48);(1,2)(3,4)(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,27,26)(22,28,31,41)(24,32,36,37)(25,33,39,40)(29,34,44,50)(30,42,46,52)(35,47,38,48)(43,49,45,51);(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24,23,32,27,36,26,37)(22,29,41,50,31,44,28,34)(25,38,33,48,39,35,40,47)(30,45,52,49,46,43,42,51);(2,4)(7,14)(10,11)(15,17)(19,20)(21,25,27,39)(22,30,31,46)(23,33,26,40)(24,35,36,38)(28,42,41,52)(29,43,44,45)(32,47,37,48)(34,49,50,51);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(24,36)(25,39)(29,44)(30,46)(32,37)(33,40)(34,50)(42,52)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7100509);