#273264

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	#62018	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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)(3,10,12,15)(4,8)(5,9)(6,11,7,13)(14,16)(17,18,23,27)(19,28,31,32)(20,30,21,25)(22,29,24,26);(1,3)(2,6)(4,13)(5,12)(7,9)(8,15)(10,16)(11,14)(17,19,23,31)(18,21,27,20)(22,25,24,30)(26,32,29,28);(4,14)(8,16)(10,15)(11,13)(17,20,23,21)(18,25,27,30)(19,29,31,26)(22,32,24,28);(1,3)(2,7)(4,13)(5,12)(6,9)(8,10)(11,14)(15,16)(17,21,23,20)(18,19,27,31)(22,32,24,28)(25,29,30,26)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,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, 180460);