#6434646

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 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)(18,19)(20,28)(21,22)(24,31)(25,26)(27,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,19,23,26,22,25)(20,24,29,27,28,32,30,31);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,21,26,23,25,22,18)(20,27,29,32,28,31,30,24);(10,14)(13,16)(17,20,23,28)(18,24,26,32)(19,27,25,31)(21,29,22,30);(1,4,5,3)(2,6,8,7)(20,28)(24,32)(27,31)(29,30)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,e^2=fg,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, 6341842);