#498358

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,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, 405554);