#273365

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	#3536	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

832 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)(19,27)(20,28)(21,26)(22,25)(23,30)(24,32)(29,31);(1,3,6,14)(2,7,10,11)(4,12,15,9)(5,13,16,8)(17,19,22,29)(18,20,25,30)(21,23,32,28)(24,31,26,27);(1,4)(2,8)(3,12)(5,7)(6,15)(9,14)(10,13)(11,16)(17,20,22,30)(18,23,25,28)(19,26,29,24)(21,27,32,31);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)(17,20,22,30)(18,19,25,29)(21,31,32,27)(23,26,28,24)

Polycyclic presentation:

2^9|a^2=fh,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, 180561);