#489481

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

657 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396677);