#200318

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107514);