#6807042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714238);