#490123

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
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

709 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(18,23)(20,31)(21,33)(24,39)(25,38)(26,42)(27,43)(28,46)(29,45)(32,35)(34,36)(37,40)(41,44)(47,48);(4,13)(8,12)(9,16)(11,14)(17,18,22,25)(19,24,30,40)(20,32,34,42)(21,35,33,26)(23,36,38,31)(27,44,28,47)(29,48,45,41)(37,46,39,43);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,19)(18,24)(20,27)(21,29)(22,30)(23,37)(25,40)(26,41)(28,34)(31,43)(32,44)(33,45)(35,48)(36,46)(38,39)(42,47);(2,7)(6,15)(8,12)(9,16)(17,20,33,36,22,34,21,31)(18,23,35,42,25,38,26,32)(19,27,45,46,30,28,29,43)(24,37,48,47,40,39,41,44)

Polycyclic presentation:

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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397319);