#322476

order	512
nilpotency class	3
isoclinism	isoclinic class of #322453
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,10,7,11,5,16)(3,12,4,15,14,9,13,8)(17,18,22,24,20,25,21,26)(19,27,30,31,29,23,28,32);(1,3,5,13,7,14,6,4)(2,8,10,12,11,15,16,9)(17,19,21,28,20,29,22,30)(18,23,26,31,25,27,24,32);(1,2)(3,8)(4,9)(5,10)(6,16)(7,11)(12,13)(14,15)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,4,5,3,7,13,6,14)(2,9,10,8,11,12,16,15)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229672);