#397113

order	512
nilpotency class	3
isoclinism	isoclinic class of #397113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14745
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304309);