#6592923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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	#32880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6500119);