#198274

order	512
nilpotency class	3
isoclinism	isoclinic class of #198117
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

1106 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105470);