#490100

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

708 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(18,24)(19,29)(20,21)(22,39)(25,43)(26,27)(28,42)(30,34)(32,33)(35,36)(37,40)(41,46)(44,48)(45,47);(4,13)(8,12)(9,16)(11,14)(17,18,23,28)(19,25,33,48)(20,24,36,42)(21,37,35,26)(22,40,39,27)(29,47,32,46)(30,45,34,41)(31,43,38,44);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,19)(18,25)(20,34)(21,38)(22,32)(23,33)(24,41)(26,43)(27,47)(28,48)(29,39)(30,36)(31,35)(37,44)(40,46)(42,45);(2,7)(6,15)(8,12)(9,16)(17,20,22,35,23,36,39,21)(18,26,27,24,28,37,40,42)(19,30,32,38,33,34,29,31)(25,44,47,45,48,43,46,41)

Polycyclic presentation:

2^9|b^2=i,d^2=g,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, 397296);