#323226

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
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

667 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:

(1,2,6,11,7,4,14,10)(3,12,5,8,13,9,16,15)(17,18,23,26,22,27,21,31)(19,28,29,25,20,24,30,32);(1,3,6,5,7,13,14,16)(2,8,10,12,4,15,11,9)(17,19,21,30,22,20,23,29)(18,24,31,25,27,28,26,32);(1,4)(2,7)(3,8)(5,12)(6,11)(9,16)(10,14)(13,15)(17,20,22,19)(18,25,27,32)(21,30,23,29)(24,26,28,31);(1,5,14,3,7,16,6,13)(2,9,11,15,4,12,10,8)(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=egh,c^2=hi,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=dhi,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, 230422);