Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,43)(34,47)(36,44)(37,45)(38,41)(39,46)(42,50)(48,52)(49,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,38,27,33,26,35)(22,28,40,43,31,41,30,32)(24,34,39,49,36,48,37,50)(29,42,45,52,44,51,46,47);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(23,33)(24,36)(28,41)(29,44)(32,43)(35,38)(37,39)(45,46);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,27,36)(22,29,31,44)(23,34,33,48)(25,39,26,37)(28,42,41,51)(30,46,40,45)(32,47,43,52)(35,50,38,49);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,35,33,38)(24,37,36,39)(28,43,41,32)(29,45,44,46)(34,49,48,50)(42,47,51,52)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434106);