#7143330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#59213	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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:

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

Polycyclic presentation:

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

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, 7050526);