#6571578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#38902	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2876 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)(22,26)(23,30)(24,32)(27,35)(28,34)(29,33);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22)(23,28)(24,33)(25,29)(26,32)(27,30)(31,34)(35,36);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,30)(22,27,29,28)(24,31,32,36)(26,34,33,35);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,23)(22,28)(24,31)(25,30)(26,35)(27,29)(32,36)(33,34);(8,16)(11,18)(14,20)(17,19)(23,30)(27,28)(31,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478774);