Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 | |
1963 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)(28,36)(33,46)(34,47)(35,41)(37,43)(38,45)(39,44)(42,49)(48,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,35,27,33,26,36)(22,28,40,46,31,41,30,32)(24,34,39,50,38,48,37,49)(29,42,43,52,45,51,44,47);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(23,33)(28,41)(32,46)(34,48)(35,36)(42,51)(47,52)(49,50);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,27,38)(22,29,31,45)(23,34,33,48)(25,39,26,37)(28,42,41,51)(30,44,40,43)(32,47,46,52)(35,50,36,49);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,35,33,36)(24,37,38,39)(28,32,41,46)(29,43,45,44)(34,49,48,50)(42,52,51,47)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=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434116);