#200955

order	512
nilpotency class	3
isoclinism	isoclinic class of #200944
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	#60962	C₂
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108151);