#107351

order	512
nilpotency class	3
isoclinism	isoclinic class of #107334
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	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,9)(3,5,8,14)(4,10)(7,12)(11,16)(13,15)(17,18,22,28,23,26,21,27)(19,24,31,38,30,39,36,42)(20,32,34,25,35,41,33,40)(29,43,45,47,44,48,46,37);(1,3,5,9,6,8,14,2)(4,10,12,16,13,15,11,7)(17,19)(18,24,26,39)(20,29)(21,31)(22,36)(23,30)(25,37,40,47)(27,38,28,42)(32,43,41,48)(33,45)(34,46)(35,44);(1,4,6,13)(2,7,9,16)(3,10,8,15)(5,12,14,11)(17,20,23,35)(18,25,26,40)(19,29,30,44)(21,34,22,33)(24,37,39,47)(27,41,28,32)(31,46,36,45)(38,48,42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=eh,f^c=fh

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, 14547);