#199411

order	512
nilpotency class	3
isoclinism	isoclinic class of #199251
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19)(18,23)(20,29)(21,28)(22,30)(24,31)(25,32)(26,27);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(2,9)(3,10)(6,13)(11,14)(18,24)(23,31)(25,26)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fi

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, 106607);