Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(22,28)(25,26)(29,41)(30,42)(31,32)(34,39)(36,45)(37,40)(43,51)(44,46)(47,52)(48,49);(7,13)(8,16)(10,17)(11,15)(21,22,26,28,27,32,25,31)(23,29,34,41,35,44,39,46)(24,36,37,45,38,42,40,30)(33,47,49,52,50,51,48,43);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,29)(24,33)(25,39)(26,34)(27,35)(28,41)(30,43)(31,46)(32,44)(36,47)(37,49)(38,50)(40,48)(42,51)(45,52);(7,13)(10,17)(14,19)(18,20)(21,24,27,38)(22,30,32,45)(23,33,35,50)(25,37,26,40)(28,42,31,36)(29,43,44,52)(34,48,39,49)(41,51,46,47);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,31,32,28)(23,34,35,39)(24,37,38,40)(29,41,44,46)(30,36,45,42)(33,48,50,49)(43,51,52,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074278);