#271912

order	512
nilpotency class	3
isoclinism	isoclinic class of #271899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

899 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179108);