#7191160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22)(23,33)(24,30)(25,32)(26,31)(27,29)(28,34)(35,36);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23,25,35,27,24,26,34)(22,28,31,30,29,36,32,33);(2,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(22,29)(23,24)(31,32)(34,35);(8,15)(11,13)(16,20)(18,19)(21,24,27,23)(22,30,29,33)(25,35,26,34)(28,32,36,31);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,27,26)(22,31,29,32)(23,34,24,35)(28,33,36,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7098356);