#107943

order	512
nilpotency class	3
isoclinism	isoclinic class of #107918
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15139);