#6432390

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	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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)(3,6)(4,5)(7,8)(9,10,13,19,15,21,14,20)(11,16,24,31,26,33,25,32)(12,18,29,17,28,36,27,35)(22,37,23,30,39,34,38,40);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,34)(18,30)(19,31)(20,32)(21,33)(22,27)(28,38)(29,39)(35,37)(36,40);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,19)(20,36)(21,35)(23,25)(24,38)(26,39)(31,37)(32,34)(33,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,29)(14,27)(15,28)(16,30)(17,19)(20,35)(21,36)(22,25)(24,39)(26,38)(31,34)(32,37)(33,40);(9,13,15,14)(10,19,21,20)(11,24,26,25)(12,27,28,29)(16,31,33,32)(17,18,35,36)(22,38,39,23)(30,37,40,34)

Polycyclic presentation:

2^9|a^2=gi,d^2=f,e^2=i,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, 6339586);