#6443970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
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	#23400	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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)(3,8)(4,6)(5,7)(9,10,16,20)(11,17,25,21)(12,14,19,15)(13,18,31,36)(22,32,39,37)(23,33,24,26)(27,29,35,30)(28,34,38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,26)(13,28)(14,23)(15,24)(16,25)(18,34)(19,33)(20,21)(22,29)(27,37)(30,39)(31,38)(32,35)(36,40);(9,12,15,10,16,19,14,20)(11,21,24,26,25,17,23,33)(13,27,30,18,31,35,29,36)(22,37,28,34,39,32,38,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,22)(12,27)(14,29)(15,30)(16,31)(17,32)(19,35)(20,36)(21,37)(23,38)(24,28)(25,39)(26,34)(33,40);(9,14,16,15)(10,19,20,12)(11,23,25,24)(13,29,31,30)(17,33,21,26)(18,35,36,27)(22,38,39,28)(32,40,37,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351166);