#117451

order	512
nilpotency class	3
isoclinism	isoclinic class of #117443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

585 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh

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, 24647);