#273348

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	#61759	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

865 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)(3,11,13,7)(4,12)(5,15,16,9)(6,10)(8,14)(17,18,24,28)(19,29,26,31)(20,25,21,30)(22,32,23,27);(1,3)(2,7)(4,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19)(18,21)(20,28)(22,30)(23,25)(24,26)(27,31)(29,32);(1,4)(2,8)(3,9)(5,11)(6,14)(7,16)(10,12)(13,15)(17,20,24,21)(18,25,28,30)(19,27,26,32)(22,31,23,29);(1,5,6,16)(2,9,10,15)(3,12,13,8)(4,11,14,7)(17,21,24,20)(18,26,28,19)(22,31,23,29)(25,32,30,27)

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^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, 180544);