#116746

order	512
nilpotency class	3
isoclinism	isoclinic class of #116738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fhi,e^c=eh,f^c=fhi,g^c=ghi

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, 23942);