#199002

order	512
nilpotency class	3
isoclinism	isoclinic class of #198825
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1060 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:

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

Polycyclic presentation:

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

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, 106198);