#197206

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	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 104402);