#6677565

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	#30519	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,25,29)(23,30,33,27)(24,34,26,28)(31,36,32,35);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,27)(24,32)(25,33)(26,31)(28,35)(29,30)(34,36);(2,4)(23,31)(24,26)(27,35)(28,34)(30,36)(32,33);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,26)(22,28,29,34)(23,32,33,31)(27,35,30,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,32)(24,26)(27,36)(28,34)(30,35)(31,33)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,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, 6584761);