#6434653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341849);