#489441

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	#62437	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

690 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:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(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);(4,13)(7,12)(9,16)(11,14)(17,19,22,32)(18,24,26,44)(20,34,21,38)(23,42,35,31)(25,45,39,48)(27,46,41,43)(28,33,47,37)(29,40,30,36);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(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);(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=h,b^2=i,d^2=gi,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, 396637);