#6456438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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

1975 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6363634);