#114550

order	512
nilpotency class	3
isoclinism	isoclinic class of #111399
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

822 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21746);