#328936

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
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	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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)(11,16)(13,15)(17,18,21,24)(19,27,28,20)(22,26,31,25)(23,32,29,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,28)(22,29)(23,31)(24,27)(25,30)(26,32);(1,3,5,2)(4,6,8,7)(10,13)(11,16)(14,15)(17,20,21,27)(18,23,24,29)(19,25,28,26)(22,32,31,30);(1,4,5,8)(2,6,3,7)(18,24)(20,27)(25,26)(30,32)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=de,f^c=fh,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236132);