#6741218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	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 ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6648414);