#201053

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

1140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=hi,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, 108249);