#490114

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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	#62432	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

694 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,22,26)(19,28,31,47)(20,33,21,38)(23,41,36,27)(24,29,43,30)(25,37,35,40)(32,48,42,45)(34,44,39,46);(4,13)(8,12)(9,16)(11,14)(17,19,22,31)(18,24,26,43)(20,34,21,39)(23,42,36,32)(25,45,35,48)(27,46,41,44)(28,33,47,38)(29,40,30,37);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,18)(19,24)(20,35)(21,25)(22,26)(23,27)(28,30)(29,47)(31,43)(32,44)(33,37)(34,48)(36,41)(38,40)(39,45)(42,46);(2,7)(6,15)(8,12)(9,16)(17,20,36,40,22,21,23,37)(18,25,41,33,26,35,27,38)(19,29,42,39,31,30,32,34)(24,28,46,48,43,47,44,45)

Polycyclic presentation:

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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397310);