#486766

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
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	#466	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×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:

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

Polycyclic presentation:

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

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, 393962);