#112220

order	512
nilpotency class	3
isoclinism	isoclinic class of #112213
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×#121	D₄
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19416);