#6432245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2182 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)(3,8)(4,6)(5,7)(9,10,13,19,15,20,14,17)(11,16,23,30,24,28,21,31)(12,18,26,34,25,33,27,32)(22,29,37,39,36,38,35,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,22)(13,23)(14,21)(15,24)(17,31)(18,29)(19,30)(20,28)(25,36)(26,37)(27,35)(32,40)(33,38)(34,39);(1,4,5,3)(2,6,7,8)(10,17)(11,21)(13,14)(16,28)(18,32)(19,20)(22,35)(23,24)(26,27)(29,38)(33,34)(36,37);(9,12)(10,18)(11,22)(13,26)(14,27)(15,25)(16,29)(17,32)(19,34)(20,33)(21,35)(23,37)(24,36)(28,38)(30,39)(31,40);(12,25)(18,33)(22,36)(26,27)(29,38)(32,34)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339441);