#323140

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230336);