#112115

order	512
nilpotency class	3
isoclinism	isoclinic class of #112109
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

857 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19311);