#490108

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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

710 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)(20,33)(21,36)(22,37)(25,39)(26,38)(27,44)(28,34)(29,30)(31,46)(35,45)(40,41)(42,48)(43,47);(4,13)(8,12)(9,16)(11,14)(17,18,23,28)(19,25,32,42)(20,34,36,24)(21,26,33,44)(22,38,37,27)(29,39,35,48)(30,43,45,40)(31,47,46,41);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,19)(18,25)(20,35)(21,30)(22,31)(23,32)(24,39)(26,43)(27,41)(28,42)(29,36)(33,45)(34,48)(37,46)(38,47)(40,44);(2,7)(6,15)(8,12)(9,16)(17,20,22,21,23,36,37,33)(18,26,27,34,28,44,38,24)(19,29,31,45,32,35,46,30)(25,40,41,39,42,43,47,48)

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=dghi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397304);