#489403

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	#62419	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

723 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 396599);