#6544658

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

1361 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6451854);