#6487384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,29)(27,36,39,28)(30,35,31,40)(33,34,37,38);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,34,32,39,31,38)(26,33,40,28,29,37,35,36);(11,17)(14,21)(18,19)(22,23)(25,28,31,33,32,36,30,37)(26,34,35,27,29,38,40,39);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,26)(27,37,39,33)(28,34,36,38)(30,35,31,40);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(25,30,32,31)(26,35,29,40)(27,38,39,34)(28,37,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394580);