#273233

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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)(3,9,11,16)(4,13,14,7)(5,10)(6,8)(12,15)(17,18)(19,27)(20,28)(21,26)(22,25)(23,30)(24,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,11)(8,13)(9,15)(10,16)(17,19,22,29)(18,20,25,30)(21,23,32,28)(24,31,26,27);(2,8)(3,11)(5,12)(9,16)(17,20,22,30)(18,23,25,28)(19,26,29,24)(21,27,32,31);(1,4,6,14)(2,9,8,16)(3,12,11,5)(7,15,13,10)(17,20,22,30)(18,19,25,29)(21,31,32,27)(23,26,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180429);