#112657

order	512
nilpotency class	3
isoclinism	isoclinic class of #112637
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

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

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, 19853);