#6532507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439703);