#6677559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22)(23,28)(24,32)(25,27)(26,30)(29,31)(33,36)(34,35);(2,4)(22,27)(23,25)(28,30)(29,32)(31,33)(35,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,23,26,25)(22,28,30,27)(24,33,34,31)(29,32,36,35);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,26,34)(22,29,30,36)(23,31,25,33)(27,32,28,35)

Polycyclic presentation:

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

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, 6584755);