#6561709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=g,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468905);