#107446

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

716 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14642);