#112001

order	512
nilpotency class	3
isoclinism	isoclinic class of #111989
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

869 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19197);