#6431966

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4694 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)(16,23)(17,29)(22,38)(24,32)(25,33)(27,34)(28,36)(30,35)(31,39)(37,40);(5,6)(7,8)(9,11,15,25)(10,16,18,32)(12,27,28,40)(13,24,14,23)(17,34,35,37)(19,33,20,21)(22,29,39,30)(26,38,36,31);(2,4)(6,8)(9,12,13,29,15,28,14,30)(10,17,19,26,18,35,20,36)(11,22,23,40,25,39,24,27)(16,31,21,37,32,38,33,34);(12,28)(17,35)(22,39)(26,36)(27,40)(29,30)(31,38)(34,37);(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,38)(34,37)

Polycyclic presentation:

2^9|b^2=i,c^2=f,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,f^b=fi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315554, #72315555, #72315588
maximal normal subgroup of	#72315554, #72315555, #72315588
maximal quotient of	#72316014, #72316026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339162);