#328356

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10078	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,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, 235552);