#273333

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3536	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

897 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)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,28)(20,25)(21,30)(22,29)(23,26)(24,27)(31,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,24,31)(18,21,27,20)(22,25,23,30)(26,32,29,28);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20,24,21)(18,25,27,30)(19,29,31,26)(22,32,23,28);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(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=h,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180529);