#7117867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=gh,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7025063);