#197290

order	512
nilpotency class	3
isoclinism	isoclinic class of #197026
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	#60805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,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, 104486);