#7187418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(9,10,14,11)(12,15,13,16)(17,18,22,26)(19,27,28,32)(20,25,21,24)(23,29,31,30);(1,3,5,8)(2,6,4,7)(10,11)(12,13)(19,28)(23,31)(27,32)(29,30);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,13,16)(17,19,20,29,22,28,21,30)(18,23,24,32,26,31,25,27);(1,4,5,2)(3,6,8,7)(9,12,14,13)(10,15,11,16);(9,13,14,12)(10,15,11,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=fgi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi

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, 7094614);