#198962

order	512
nilpotency class	3
isoclinism	isoclinic class of #198798
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

1088 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:

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

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72238925, #72238930, #72238944
maximal normal subgroup of	#72238925, #72238930, #72238944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106158);