#241020

order	512
nilpotency class	3
isoclinism	isoclinic class of #241009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,30)(28,31)(29,32);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,19,23,31)(18,24,26,32)(20,27,25,28)(21,30,22,29);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(17,20,22,18,23,25,21,26)(19,28,30,24,31,27,29,32);(3,11)(7,14)(10,16)(13,15)(17,21,23,22)(18,25,26,20)(19,29,31,30)(24,27,32,28)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148216);