#200982

order	512
nilpotency class	3
isoclinism	isoclinic class of #200920
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

1108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,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, 108178);