#323135

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
prime radical	#19629	C₂
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)(9,10,14,19,15,12,20,18)(11,17,21,16,22,24,13,23);(1,3,5,2,4,8,6,7)(9,11,20,13,15,22,14,21)(10,16,19,24,12,23,18,17);(2,3)(5,6)(7,8)(9,12)(10,15)(11,16)(13,24)(14,19)(17,21)(18,20)(22,23);(2,7)(3,8)(9,13,20,22,15,21,14,11)(10,17,19,16,12,24,18,23)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=ei,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, 230331);