#489527

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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	#62488	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396723);