#489404

order	512
nilpotency class	4
isoclinism	isoclinic class of #489355
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
prime radical	#62420	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396600);