#7160937

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=h,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=dh,e^c=ei,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, 7068133);