#489610

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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	#62344	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396806);