#6542428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533063
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
prime radical	#62955	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449624);