#6806836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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
prime radical	#60763	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,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,e^c=ei,g^c=gi

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, 6714032);