#6621379

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(9,10,13,15,12,16,18,22)(11,17,20,24,19,14,21,23);(2,6)(4,7)(9,11,18,20,12,19,13,21)(10,14,15,24,16,17,22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,22)(15,18)(17,19)(20,24)(21,23);(1,3,5,8)(2,7,6,4)(11,19)(14,17)(20,21)(23,24);(1,4,5,7)(2,3,6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72321053

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528575);