#513219

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
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

381 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)(10,15)(11,19)(12,21)(13,25)(16,28)(17,31)(18,32)(20,24)(22,38)(23,35)(26,37)(27,29)(30,39)(33,34)(36,40);(1,3)(2,5)(4,8)(6,7)(10,16)(15,28)(17,30)(18,27)(20,36)(24,40)(29,32)(31,39);(9,10)(11,20)(12,24)(13,27)(14,16)(15,19)(17,21)(18,25)(22,39)(23,36)(26,40)(28,35)(29,38)(30,37)(31,33)(32,34);(3,8)(5,7)

Polycyclic presentation:

2^9|a^2=e,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	#72234321, #72322380, #72322387, …
maximal normal subgroup of	#72234321, #72322380, #72322387, …
maximal quotient of	#72322872, #72322888

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420415);