#119069

order	512
nilpotency class	3
isoclinism	isoclinic class of #119068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

500 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)(3,4,8,14)(5,9)(6,10)(7,12,13,15)(11,16)(17,18)(19,27,29,23)(20,24,32,41)(21,34)(22,25)(26,37)(28,38)(30,39,35,43)(31,48)(33,44,36,42)(40,47)(45,46);(1,3)(2,7)(4,11)(5,15)(6,8)(9,14)(10,13)(12,16)(17,19,22,29)(18,23,25,27)(20,31,32,47)(21,35,37,30)(24,40,41,48)(26,43,34,39)(28,36,46,33)(38,44,45,42);(1,4,5,13,6,14,16,7)(2,8,9,15,10,3,11,12)(17,20)(18,24,25,41)(19,28)(21,36)(22,32)(23,38,27,45)(26,44,34,42)(29,46)(30,47)(31,35)(33,37)(39,48,43,40)

Polycyclic presentation:

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

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, 26265);