#323141

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=e,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, 230337);