#201951

order	512
nilpotency class	3
isoclinism	isoclinic class of #201948
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60750	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1287 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,23,26)(19,21,25,22)(20,27)(24,28)(29,31)(30,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,16)(12,15)(17,19,22,18,23,25,21,26)(20,27,29,24,30,32,28,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31);(2,8)(6,14)(7,11)(13,16)(17,21,23,22)(18,19,26,25)(20,28,30,29)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109147);