#198223

order	512
nilpotency class	3
isoclinism	isoclinic class of #198144
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

1057 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,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, 105419);