#289304

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

543 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efh,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196500);