#119072

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

549 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=h,g^2=hi,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, 26268);