#489777

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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	#61284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

691 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)(7,12)(13,16)(14,15)(17,18,23,28)(19,25,31,45)(20,26,22,38)(21,34,37,48)(24,39,42,30)(27,33,46,36)(29,43,32,44)(35,40,47,41);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,15,10,11)(17,19)(18,24)(20,29)(21,35)(22,32)(23,31)(25,34)(26,40)(27,44)(28,42)(30,36)(33,39)(37,47)(38,41)(43,46)(45,48);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,25)(20,32)(21,30)(22,29)(23,31)(24,34)(26,44)(27,40)(28,45)(33,47)(35,36)(37,39)(38,43)(41,46)(42,48);(2,9)(7,16)(8,10)(11,15)(17,20,23,22)(18,26,28,38)(19,29,31,32)(21,33,37,36)(24,40,42,41)(25,43,45,44)(27,34,46,48)(30,35,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396973);