#329354

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=eh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236550);