#6544600

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	#23573	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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)(27,35)(28,37)(29,32)(30,33)(31,34)(36,39)(38,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27)(26,28)(29,37)(30,36)(31,38)(32,40)(33,35)(34,39);(11,18)(14,17)(19,24)(21,23)(25,28)(26,27)(29,35)(30,40)(31,39)(32,36)(33,37)(34,38);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(25,29,31,30)(26,32,34,33)(27,36,38,37)(28,35,39,40);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,31,30)(26,33,34,32)(27,37,38,36)(28,35,39,40)

Polycyclic presentation:

2^9|a^2=fgi,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, 6451796);