#200523

order	512
nilpotency class	3
isoclinism	isoclinic class of #200400
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,22,26,23,25,21,19)(20,28,29,32,30,24,31,27);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,19,22,25,23,26,21,18)(20,27,31,28,30,32,29,24);(17,20,23,30)(18,24,25,28)(19,27,26,32)(21,29,22,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,21,23,22)(18,19,25,26)(20,29,30,31)(24,27,28,32)

Polycyclic presentation:

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

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, 107719);