#489329

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	#61502	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

679 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,24,20)(19,28,32,46)(21,26,23,33)(22,36,39,47)(25,41,29,31)(27,35,34,38)(30,45,40,44)(37,42,48,43);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,25)(20,29)(21,30)(22,37)(23,40)(24,32)(26,42)(27,44)(28,36)(31,38)(33,43)(34,45)(35,41)(39,48)(46,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20,24,18)(19,29,32,25)(21,26,23,33)(22,27,39,34)(28,41,46,31)(30,42,40,43)(35,47,38,36)(37,44,48,45);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,21,24,23)(18,26,20,33)(19,30,32,40)(22,35,39,38)(25,42,29,43)(27,36,34,47)(28,45,46,44)(31,37,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

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, 396525);