#317954

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	#19332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

548 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)(3,10)(4,6)(5,9)(7,12)(8,11)(13,14)(15,16)(18,22)(19,26)(20,23)(25,31)(28,29)(30,32);(2,6,9,8)(7,14,15,10)(17,18,23,31,21,25,20,22)(19,27,26,30,29,24,28,32);(1,2)(3,7)(4,8)(5,9)(6,11)(10,12)(13,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,27)(31,32);(1,3,11,16,5,13,4,12)(2,7,6,14,9,15,8,10)(17,18,23,31,21,25,20,22)(19,24,26,32,29,27,28,30)

Polycyclic presentation:

2^9|b^2=eghi,d^2=eh,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, 225150);