#322388

order	512
nilpotency class	3
isoclinism	isoclinic class of #322377
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	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

725 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egi,d^2=eh,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=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229584);