#6618370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×AGL(1,ℤ/8)	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,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, 6525566);