#6489537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396733);