#7083042

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,30)(22,24)(23,28)(27,31)(29,32);(9,10)(11,15)(12,14)(13,16)(17,19,21,28,22,27,20,29)(18,23,26,25,24,32,30,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(18,24)(19,27)(26,30)(28,29);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(17,19)(18,25)(20,28)(21,29)(22,27)(23,26)(24,31)(30,32);(10,14)(13,15)(17,20,22,21)(18,26,24,30)(19,28,27,29)(23,31,32,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=gh,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 6990238);