#256500

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45991	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163696);