#6566046

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,37)(28,39)(29,34)(30,35)(31,36)(32,40)(33,38);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,27)(26,33)(28,32)(29,30)(31,38)(34,40)(35,39)(36,37);(11,18)(14,17)(19,24)(21,23)(25,28,31,29)(26,33,36,37)(27,30,38,32)(34,40,39,35);(1,4,5,3)(2,6,7,8)(12,20)(15,22)(19,24)(21,23)(25,29,31,28)(26,34,36,39)(27,30,38,32)(33,40,37,35);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,30,31,32)(26,35,36,40)(27,28,38,29)(33,34,37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,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, 6473242);