#491532

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 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)(20,34)(21,26)(22,31)(23,29)(24,27)(25,33)(28,37)(32,43)(35,44)(36,42)(38,46)(39,48)(40,47)(41,45);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,21,33,23,35,24,32)(18,25,27,30,29,43,26,44)(20,36,38,45,39,31,40,28)(22,34,41,47,42,48,37,46);(3,10)(6,15)(9,16)(11,14)(17,20,37,25,24,40,42,44,23,39,41,43,21,38,22,30)(18,19,31,46,26,32,45,48,29,35,36,47,27,33,28,34);(17,21,23,24)(18,26,29,27)(19,32,35,33)(20,38,39,40)(22,41,42,37)(25,30,43,44)(28,31,45,36)(34,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398728);