#489327

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	#61501	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

682 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,22,27)(19,28,31,46)(20,25,23,32)(21,34,37,47)(24,40,43,30)(26,33,36,38)(29,45,39,44)(35,41,48,42);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,29)(21,35)(22,31)(23,39)(25,41)(26,44)(27,43)(28,34)(30,38)(32,42)(33,40)(36,45)(37,48)(46,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,18)(19,24)(20,32)(21,36)(22,27)(23,25)(26,37)(28,40)(29,42)(30,46)(31,43)(33,34)(35,45)(38,47)(39,41)(44,48);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,20,22,23)(18,25,27,32)(19,29,31,39)(21,33,37,38)(24,41,43,42)(26,34,36,47)(28,45,46,44)(30,35,40,48)

Polycyclic presentation:

2^9|a^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, 396523);