#234348

order	512
nilpotency class	3
isoclinism	isoclinic class of #234343
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1311 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)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,29)(18,23,26,31)(20,28,21,30)(24,32,25,27);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141544);