#6432362

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	#23439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6339558);