#6557767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,19)(12,21)(13,17)(14,16)(15,18)(20,23)(22,24)(26,32)(27,36)(29,30)(33,37)(34,35)(39,40);(2,5)(6,8)(9,11,15,14)(10,16,18,19)(12,22,13,20)(17,24,21,23)(25,26,29,32,31,34,30,35)(27,28,33,39,37,38,36,40);(1,3,4,7)(2,6,5,8)(9,12,15,13)(10,17,18,21)(11,20,14,22)(16,23,19,24)(25,27,30,36,31,37,29,33)(26,28,35,40,34,38,32,39);(9,13,15,12)(10,17,18,21)(11,20,14,22)(16,24,19,23)(25,28,31,38)(26,33,34,36)(27,32,37,35)(29,39,30,40);(9,14,15,11)(10,16,18,19)(12,22,13,20)(17,23,21,24)(26,34)(28,38)(32,35)(39,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464963);