#6544168

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2793 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)(21,22,27,29,26,23,25,30)(24,31,32,35,34,28,33,36);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,22)(24,31,34,28)(25,30,27,29)(32,36,33,35);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,36)(30,35);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,29,23,30)(24,32,34,33)(28,35,31,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6451364);