#6572580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	#36411	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2931 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)(21,22,25,30,26,28,24,29)(23,31,32,35,33,36,34,27);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23)(22,27)(24,32)(25,34)(26,33)(28,35)(29,31)(30,36);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,28)(27,35)(29,30)(31,36);(8,16)(11,18)(14,20)(17,19)(21,24,26,25)(22,29,28,30)(23,32,33,34)(27,31,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479776);