#494194

order	512
nilpotency class	4
isoclinism	isoclinic class of #483399
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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)(3,6)(4,5)(7,8)(11,19,20,31)(12,22)(13,24)(15,25,26,36)(17,30)(18,29)(21,34,33,32)(23,35)(27,39,38,37)(28,40);(2,5)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,27)(22,30)(23,28)(24,29)(31,36)(32,37)(33,38)(34,39)(35,40);(9,11,12,19,13,21,22,32,14,20,23,31,24,33,35,34)(10,15,17,25,18,27,30,37,16,26,28,36,29,38,40,39);(1,3,4,7)(2,6,5,8)(10,16)(15,26)(17,28)(18,29)(25,36)(27,38)(30,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313603, #72313642, #72313668, …
maximal normal subgroup of	#72313603, #72313642, #72313668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401390);