#489568

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

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, 396764);