#205590

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112786);