#198239

order	512
nilpotency class	3
isoclinism	isoclinic class of #198162
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	C₂×#447	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 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:

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

Polycyclic presentation:

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

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, 105435);