#7187638

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1276 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)(10,14)(12,15)(17,18)(19,25)(20,29)(21,23)(22,28)(24,30)(26,32)(27,31);(9,10,13,14)(11,12,16,15)(17,19,21,26)(18,22,23,31)(20,27,30,28)(24,32,29,25);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,14)(13,16)(18,23)(19,26)(24,29)(27,28);(10,14)(11,16)(18,23)(19,27)(20,30)(22,25)(26,28)(31,32);(9,12,13,15)(10,11,14,16)(18,23)(19,26)(24,29)(27,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi

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, 7094834);