#199461

order	512
nilpotency class	3
isoclinism	isoclinic class of #199203
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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:

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106657);