#197403

order	512
nilpotency class	3
isoclinism	isoclinic class of #197348
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	#37556	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1154 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:

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

Polycyclic presentation:

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

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, 104599);