#197211

order	512
nilpotency class	3
isoclinism	isoclinic class of #197038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 104407);