#289393

order	512
nilpotency class	3
isoclinism	isoclinic class of #289363
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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

555 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196589);