#513223

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	D₄×D₁₆	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

385 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234374, #72322373, #72322434
maximal normal subgroup of	#72234374, #72322373, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420419);