#489682

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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	#62383	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

637 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,24,20)(19,28,32,47)(21,36,22,40)(23,35,38,27)(25,30,29,31)(26,39,34,42)(33,48,43,45)(37,44,41,46);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,25)(20,29)(21,37)(22,41)(23,43)(24,32)(26,45)(27,46)(28,40)(30,39)(31,42)(33,38)(34,48)(35,44)(36,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20,24,18)(19,29,32,25)(21,34,22,26)(23,35,38,27)(28,30,47,31)(33,46,43,44)(36,42,40,39)(37,48,41,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,38,42,24,22,23,39)(18,26,35,36,20,34,27,40)(19,30,43,41,32,31,33,37)(25,28,46,48,29,47,44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396878);