#7014322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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

1341 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)(9,10)(11,13)(12,14)(15,16)(17,18,20,25,21,22,30,24)(19,26,29,31,28,32,27,23);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,22)(23,31)(24,25)(26,32);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,28)(18,23,22,31)(20,27,30,29)(24,26,25,32);(3,8)(6,7)(11,16)(13,15)(17,20,21,30)(18,24,22,25)(19,27,28,29)(23,26,31,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6921518);