#489422

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	#62439	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,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, 396618);