Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2700 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)(21,22,26,29)(23,31,34,44)(24,28,35,42)(25,37,39,30)(27,38,41,33)(32,45,46,52)(36,49,48,43)(40,50,51,47);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,34)(22,27,29,41)(24,32,35,46)(25,38,39,33)(28,40,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,26,35)(22,28,29,42)(23,32,34,46)(25,36,39,48)(27,40,41,51)(30,43,37,49)(31,45,44,52)(33,47,38,50);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(45,52)(47,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,29)(27,41)(28,42)(30,37)(31,44)(40,51)(43,49)(45,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442311);