#489472

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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	#62488	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,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,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396668);