#496368

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1130 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)(4,6)(5,8)(9,10,12,16,14,19,26,18)(11,20,23,34,24,37,21,36)(13,27,25,17,29,33,38,35)(15,28,31,22,32,39,30,40);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,15)(12,21)(13,28)(14,24)(16,30)(17,34)(18,31)(19,32)(20,33)(22,38)(23,26)(25,40)(27,37)(29,39)(35,36);(2,6)(5,7)(9,12,14,26)(10,16,19,18)(11,21,24,23)(13,25,29,38)(15,30,32,31)(17,33,35,27)(20,36,37,34)(22,28,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310135, #72310144, #72310148
maximal normal subgroup of	#72310135, #72310144, #72310148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403564);