#119000

order	512
nilpotency class	3
isoclinism	isoclinic class of #118997
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

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, 26196);