#6454400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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)(9,10,13,18,14,19,21,17)(11,20,12,16,22,15,23,24);(9,10)(11,16)(12,20)(13,17)(14,19)(15,23)(18,21)(22,24);(2,5)(6,8)(9,11,14,22)(10,15,19,20)(12,21,23,13)(16,18,24,17);(9,12)(10,16)(11,21)(13,22)(14,23)(15,18)(17,20)(19,24);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,19,18)(11,12,22,23)(15,16,20,24)

Polycyclic presentation:

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

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, 6361596);