#7150058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#59040	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7057254);