#112498

order	512
nilpotency class	3
isoclinism	isoclinic class of #112491
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

791 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:

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

Polycyclic presentation:

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

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, 19694);