#6687259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#37699	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594455);