#6819942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1182 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)(3,6)(4,8)(5,7)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,26,30,35)(27,33,37,40)(28,32,38,34)(29,39,36,31);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,28,36,30,37,38,29)(26,31,34,33,35,39,32,40);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28,30,38)(26,32,35,34)(27,36,37,29)(31,40,39,33);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,38,37,30,36,28,27)(26,33,32,31,35,40,34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727138);