Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 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)(21,22,26,30)(23,31,36,48)(24,28,37,44)(25,39,32,29)(27,40,43,35)(33,46,34,47)(38,52,49,45)(41,51,42,50);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,32,40,26,36,25,35)(22,27,39,48,30,43,29,31)(24,33,49,51,37,34,38,50)(28,41,52,47,44,42,45,46);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,28)(23,33)(25,38)(26,37)(27,41)(29,45)(30,44)(31,46)(32,49)(34,36)(35,50)(39,52)(40,51)(42,43)(47,48);(7,14)(10,18)(13,15)(17,19)(21,24)(22,28)(23,34)(25,38)(26,37)(27,42)(29,45)(30,44)(31,47)(32,49)(33,36)(35,51)(39,52)(40,50)(41,43)(46,48);(6,12)(10,18)(11,20)(17,19)(24,37)(28,44)(33,34)(38,49)(41,42)(45,52)(46,47)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384941);