#6423637

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2408	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,22,24,21,23,19,26)(20,28,30,32,29,25,27,31);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,15)(13,14)(18,23)(24,26)(25,28)(31,32);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,19,21,22)(18,24,23,26)(20,27,29,30)(25,31,28,32);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20,19,27,21,29,22,30)(18,25,26,32,23,28,24,31);(9,12,13,11)(10,15,14,16)(17,19,21,22)(18,24,23,26)(20,27,29,30)(25,31,28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343555, #72343871, #72343887
maximal normal subgroup of	#72343555, #72343871, #72343887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6330833);