#7512490

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
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	#34964	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281225, #72282054
maximal normal subgroup of	#72281225, #72282054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419686);