#6489154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25729	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,25,22,24,20,29)(19,26,27,31,28,23,30,32);(11,12)(14,15)(17,18)(19,26)(20,25)(21,29)(22,24)(23,28)(27,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,28)(18,23,24,26)(20,30,21,27)(25,32,29,31);(9,12,13,11)(10,15,16,14)(18,24)(23,26)(25,29)(31,32);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396350);