#7145540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45423	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 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)(3,6)(4,5)(7,8)(11,15)(14,16)(18,22)(19,23)(24,27)(25,30)(26,29)(31,32);(2,5)(3,7)(10,13)(11,15)(17,18,21,22)(19,23,26,29)(20,24,28,27)(25,30,32,31);(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31);(9,11,12,15)(10,14,13,16)(17,18,21,22)(19,23,26,29)(20,27,28,24)(25,31,32,30)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052736);