#6677239

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	#29662	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(22,27)(23,24)(30,31)(33,34);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,25,27)(23,32,24,29)(26,30,35,31)(28,34,36,33);(2,4)(21,23,25,24)(22,28,27,36)(26,34,35,33)(29,30,32,31);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,23)(22,29,27,32)(26,34,35,33)(28,30,36,31);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,30)(26,35)(27,31)(28,32)(29,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584435);