#490166

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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

656 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)(3,12)(4,13)(5,9)(6,10)(7,11)(8,15)(14,16)(17,18)(19,31)(20,24)(21,29)(22,28)(23,30)(25,33)(26,37)(27,35)(32,42)(34,45)(36,43)(38,41)(39,44)(40,48)(46,47);(1,3,7,8)(2,4,11,16)(5,14,6,13)(9,12,10,15)(18,24)(20,35)(21,22)(25,40)(26,28)(27,29)(30,41)(31,42)(32,47)(33,39)(36,48)(37,38)(43,45)(44,46);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,19)(18,25)(20,36)(21,39)(22,33)(23,34)(24,40)(26,43)(27,42)(28,45)(29,31)(30,44)(32,38)(35,48)(37,47)(41,46);(1,5,7,6)(2,9,11,10)(3,13,8,14)(4,15,16,12)(17,20,21,37,23,38,22,35)(18,26,28,24,30,27,29,41)(19,32,39,48,34,36,33,47)(25,42,45,46,44,43,31,40)

Polycyclic presentation:

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

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, 397362);