#118310

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

609 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:

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

Polycyclic presentation:

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

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, 25506);