#112474

order	512
nilpotency class	3
isoclinism	isoclinic class of #112473
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

765 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18)(19,28,31,24)(20,32)(21,37)(22,27)(23,26)(25,34)(29,42,46,48)(30,40,38,41)(33,45,47,39)(35,43)(36,44);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,28)(27,40)(29,35)(32,39)(34,46)(36,47)(37,41)(43,48)(44,45);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(17,20,23,36)(18,25,26,43)(19,29,31,46)(21,35,22,34)(24,39,28,45)(27,44,37,32)(30,47,38,33)(40,42,41,48)

Polycyclic presentation:

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

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, 19670);