#6454852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3091 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)(3,6)(4,5)(7,8)(9,10,16,20)(11,14,19,15)(12,17,13,18)(21,23,22,24);(9,11,15,10,16,19,14,20)(12,21,23,18,13,22,24,17);(2,5)(6,8)(9,12,16,13)(10,17,20,18)(11,21,19,22)(14,23,15,24);(9,13,16,12)(10,18,20,17)(11,22,19,21)(14,24,15,23);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,11,20,19)(12,23,13,24)(17,21,18,22)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=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, 6362048);