#273354

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

912 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)(3,9,11,7)(4,6)(5,12,13,8)(10,14)(15,16)(17,18)(19,28)(20,30)(21,24)(22,27)(23,26)(25,32)(29,31);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19)(18,21)(20,26)(22,30)(23,29)(24,32)(25,31)(27,28);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20)(18,24)(19,27)(21,23)(22,31)(25,29)(26,30)(28,32);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21)(18,19)(20,23)(22,31)(24,27)(25,30)(26,29)(28,32)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=hi,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, 180550);