#197308

order	512
nilpotency class	3
isoclinism	isoclinic class of #197038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1102 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)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18,21,25,22,27,20,26)(19,24,28,23,30,31,29,32);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,31)(27,32);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,19)(18,24)(20,29)(21,28)(22,30)(23,25)(26,32)(27,31);(2,8)(3,10)(9,14)(11,12)(17,20,22,21)(18,25,27,26)(19,28,30,29)(23,24,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,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, 104504);