#489226

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

695 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)(19,28)(20,25)(21,34)(22,32)(23,27)(24,30)(26,36)(29,45)(31,46)(33,43)(35,39)(37,44)(38,42)(40,48)(41,47);(4,13)(7,12)(9,16)(11,14)(17,19,23,31)(18,24,27,41)(20,29,22,38)(21,35,37,48)(25,39,32,40)(26,42,43,45)(28,44,46,34)(30,33,47,36);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,18)(19,24)(20,32)(21,26)(22,25)(23,27)(28,30)(29,40)(31,41)(33,44)(34,36)(35,42)(37,43)(38,39)(45,48)(46,47);(17,20,23,22)(18,25,27,32)(19,29,31,38)(21,33,37,36)(24,39,41,40)(26,34,43,44)(28,45,46,42)(30,35,47,48)

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396422);