#6572221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(3,7)(4,8)(5,6)(25,26,30,34)(27,29,33,31)(28,35,38,36)(32,39,40,37);(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27)(26,31)(28,36)(29,34)(30,33)(32,37)(35,38)(39,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,30,38)(26,32,34,40)(27,35,33,36)(29,37,31,39);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,26)(27,29)(28,32)(30,34)(31,33)(35,37)(36,39)(38,40);(1,4,5,3)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,36,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,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, 6479417);