#6612001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#24384	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,25,22,19,21,26)(20,24,30,31,28,27,29,32);(9,11,13,12)(10,14,15,16)(17,19,22,18)(20,27,28,24)(21,26,23,25)(29,32,30,31);(2,5)(6,8)(10,15)(11,12)(17,20,22,28)(18,24,19,27)(21,29,23,30)(25,31,26,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(20,28)(24,27)(29,30)(31,32);(11,12)(14,16)(17,21,22,23)(18,25,19,26)(20,29,28,30)(24,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283396, #72283463, #72283591, …
maximal normal subgroup of	#72283396, #72283463, #72283591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519197);