Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 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)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,32)(24,30)(25,31)(26,40)(27,29)(28,35)(33,47)(34,41)(36,46)(37,43)(38,44)(39,45)(42,49)(48,52)(50,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,25,34,27,36,26,35)(22,28,40,46,29,41,31,32)(24,33,39,50,37,48,38,49)(30,42,44,52,43,51,45,47);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,29)(24,37)(28,41)(31,40)(32,46)(33,48)(38,39)(49,50);(8,16)(11,20)(13,15)(17,19)(21,24,27,37)(22,30,29,43)(23,33,36,48)(25,39,26,38)(28,42,41,51)(31,45,40,44)(32,47,46,52)(34,50,35,49);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,31,29,40)(23,34,36,35)(24,38,37,39)(28,32,41,46)(30,44,43,45)(33,49,48,50)(42,52,51,47)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418087);