#111997

order	512
nilpotency class	3
isoclinism	isoclinic class of #111989
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

890 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)(3,10,12,6)(4,9)(5,8)(7,11,15,13)(14,16)(17,18,21,24,22,26,20,25)(19,27,28,23,30,32,29,31);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,19,22,30)(18,23,26,31)(20,29,21,28)(24,32,25,27);(1,3)(2,7)(4,13)(5,12)(6,9)(8,15)(10,16)(11,14)(18,24)(20,21)(23,27)(25,26)(28,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19193);