Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,27)(23,31,33,44)(24,35,36,29)(26,39,38,30)(28,40,41,34)(32,46,47,45)(37,43,51,50)(42,48,49,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,27)(28,41)(29,35)(30,39)(31,44)(42,49)(43,50)(45,46);(2,4)(21,23,25,33)(22,28,27,41)(24,32,36,47)(26,40,38,34)(29,42,35,49)(30,44,39,31)(37,52,51,48)(43,46,50,45);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,32)(25,36)(26,37)(27,35)(28,42)(30,43)(31,45)(33,47)(34,48)(38,51)(39,50)(40,52)(41,49)(44,46);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,36)(29,35)(32,47)(37,51)(42,49)(43,50)(45,46)(48,52)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580989);