#6447237

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,28,31)(20,24,21,25)(27,30,29,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(1,4,5,3)(2,6,8,7)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316917, #72316919, #72316960
maximal normal subgroup of	#72316917, #72316919, #72316960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354433);