#273297

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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

812 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)(3,11,12,9)(4,14,15,7)(5,10)(6,8)(13,16)(17,18,22,26)(19,27,29,30)(20,23,31,28)(21,25,32,24);(1,3)(2,7)(4,13)(5,15)(6,12)(8,14)(9,10)(11,16)(17,19,22,29)(18,20,26,31)(21,23,32,28)(24,30,25,27);(2,8)(3,12)(5,13)(9,11)(17,20,22,31)(18,23,26,28)(19,25,29,24)(21,27,32,30);(1,4,6,15)(2,9,8,11)(3,13,12,5)(7,10,14,16)(17,20,22,31)(18,19,26,29)(21,30,32,27)(23,25,28,24)

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=dei,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180493);