#6535020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2774 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)(23,29)(24,26)(25,33)(27,31)(28,35)(30,34)(32,36);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,32)(22,27,24,30)(25,34,35,31)(28,36,33,29);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,26,22)(23,30,32,27)(25,33,35,28)(29,31,36,34);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,24)(27,30)(28,33)(29,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^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^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, 6442216);