#486781

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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	C₂²×C₈	C₂⁴
prime radical	#12164	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393977);