#273856

order	512
nilpotency class	3
isoclinism	isoclinic class of #273819
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	#61200	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

909 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=hi,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,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

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, 181052);