#6458802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437412
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2620 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)(19,27)(20,25)(21,26)(22,24)(23,28)(29,32)(30,31);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,24)(23,31)(25,26)(27,32);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,32,31,27);(17,20,22,21)(18,26,24,25)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281369, #72281492, #72281513, …
maximal normal subgroup of	#72281369, #72281492, #72281513, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365998);