#317951

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

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, 225147);