Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6447057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2420 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14)(4,19)(5,11)(6,29)(7,12)(8,13)(9,17)(10,22)(15,35)(16,42)(18,33)(20,41)(21,46)(23,37)(24,47)(25,54)(26,39)(27,56)(28,40)(30,34)(31,49)(32,45)(36,52)(38,55)(43,59)(44,58)(48,60)(50,64)(51,62)(53,61)(57,63);(1,3,6,18,8,17,7,30)(2,9,12,33,13,14,29,34)(4,20,22,44,24,49,23,15)(5,25,27,45,26,55,28,16)(10,35,19,31,37,58,47,41)(11,38,40,32,39,54,56,42)(21,50,52,63,51,43,53,48)(36,60,46,59,61,57,62,64);(1,4,6,22,8,24,7,23)(2,10,12,19,13,37,29,47)(3,15,30,49,17,44,18,20)(5,21,27,52,26,51,28,53)(9,31,34,35,14,41,33,58)(11,36,40,46,39,61,56,62)(16,43,55,63,45,50,25,48)(32,57,38,59,42,60,54,64);(1,5)(2,11)(3,16,17,45)(4,21)(6,27)(7,28)(8,26)(9,32,14,42)(10,36)(12,40)(13,39)(15,43,44,50)(18,25,30,55)(19,46)(20,48,49,63)(22,52)(23,53)(24,51)(29,56)(31,57,41,60)(33,54,34,38)(35,59,58,64)(37,61)(47,62);(5,26)(11,39)(16,45)(21,51)(25,55)(27,28)(32,42)(36,61)(38,54)(40,56)(43,50)(46,62)(48,63)(52,53)(57,60)(59,64)Polycyclic presentation:
2^9|b^2=g,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354253);