#110547

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efhi,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17743);