Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2856 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,30)(23,33,37,44)(24,38,41,46)(25,31,26,32)(28,36,43,35)(29,40,45,39)(34,48,51,42)(47,50,52,49);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,37)(22,28,30,43)(24,34,41,51)(25,36,26,35)(29,42,45,48)(31,33,32,44)(38,52,46,47)(39,50,40,49);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,45)(31,46)(32,38)(33,47)(35,49)(36,50)(37,51)(43,48)(44,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(28,43)(29,45)(31,32)(33,44)(38,46)(42,48)(47,52);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,30,32)(23,35,37,36)(24,39,41,40)(28,44,43,33)(29,46,45,38)(34,49,51,50)(42,52,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388895);