#7167008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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

1777 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,30)(23,31,34,35)(24,25,29,26)(28,36,32,33);(7,13)(8,16)(10,17)(11,15)(21,23,25,36,27,34,26,33)(22,28,24,35,30,32,29,31);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19);(7,13)(10,17)(14,19)(18,20)(21,24)(22,25)(23,32)(26,30)(27,29)(28,34)(31,33)(35,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,29,30,24)(23,33,34,36)(28,35,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7074204);