#370585

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 277781);