#7013873

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6921069);