#197040

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	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=h,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, 104236);