#499570

order	512
nilpotency class	4
isoclinism	isoclinic class of #498966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265585, #72265599, #72265621
maximal normal subgroup of	#72265585, #72265599, #72265621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406766);