#273419

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
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	C₂³×C₄	C₂⁴
prime radical	#3537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

849 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)(3,11,14,7)(4,13)(5,12,16,9)(6,10)(8,15)(17,18,23,27)(19,29,25,31)(20,30,21,24)(22,28,32,26);(1,3,6,14)(2,7,10,11)(4,12,15,9)(5,13,16,8)(17,19,23,25)(18,21,27,20)(22,24,32,30)(26,31,28,29);(1,4)(2,8)(3,12)(5,7)(6,15)(9,14)(10,13)(11,16)(17,20)(18,24)(19,26)(21,23)(22,31)(25,28)(27,30)(29,32);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)(17,21)(18,25)(19,27)(20,23)(22,31)(24,28)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,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, 180615);