#6432141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=di,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316135, #72316163, #72316171
maximal normal subgroup of	#72316135, #72316163, #72316171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339337);