#6555054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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

1319 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=fghi,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^d=eh

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, 6462250);