#253790

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160986);