#6432388

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

Polycyclic presentation:

2^9|a^2=g,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, 6339584);