#105889

order	512
nilpotency class	3
isoclinism	isoclinic class of #105878
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

804 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:

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

Polycyclic presentation:

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

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, 13085);