#197255

order	512
nilpotency class	3
isoclinism	isoclinic class of #197002
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1149 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)(3,12,4,8)(5,15)(6,11)(7,10)(9,14,16,13)(17,18,21,24,22,26,20,25)(19,27,28,23,30,31,29,32);(1,3,6,4)(2,8)(5,13,7,14)(9,10)(11,12)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,31)(26,32);(1,4,6,3)(2,9,11,16)(5,14,7,13)(8,10,12,15);(1,5,6,7)(2,10,11,15)(3,13,4,14)(8,9,12,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=gi,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, 104451);