#118930

order	512
nilpotency class	3
isoclinism	isoclinic class of #118926
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

551 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:

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

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=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=gi

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, 26126);