#6469039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24214	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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)(11,15)(13,16)(17,18,23,28)(19,29,21,26)(20,27,22,25)(24,30,32,31);(9,10)(11,16)(12,14)(13,15)(17,19)(18,24)(20,31)(21,23)(22,30)(25,29)(26,27)(28,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20,23,22)(18,25,28,27)(19,30,21,31)(24,26,32,29);(10,14)(13,16)(17,21,23,19)(18,26,28,29)(20,31,22,30)(24,25,32,27);(1,4,5,3)(2,6,7,8)(17,20,23,22)(18,27,28,25)(19,31,21,30)(24,26,32,29)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376235);