#7088172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(3,6,4,8)(9,10)(11,13)(12,14)(15,16)(18,23)(20,21)(24,29)(25,31)(26,32)(28,30);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,26,27,31)(20,23,21,29)(25,30,32,28);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(18,24)(19,27)(23,29)(28,30);(11,16)(13,15)(17,19)(18,25)(20,28)(21,30)(22,27)(23,31)(24,32)(26,29);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,20,22,21)(18,23,24,29)(19,28,27,30)(25,31,32,26)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,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, 6995368);