#198667

order	512
nilpotency class	3
isoclinism	isoclinic class of #198522
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105863);