#6355528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi

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, 6262724);