#491864

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3283	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 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)(3,9,12,15)(4,13,14,7)(6,10,16,11)(17,18,22,26)(19,27,28,45)(20,31,35,47)(21,36,33,25)(23,38,39,29)(24,41,43,34)(30,46,37,42)(32,44,48,40);(1,3,5,12)(2,6,8,16)(4,10,14,11)(7,15,13,9);(17,19)(18,23)(20,32)(21,37)(22,28)(24,42)(25,44)(26,39)(27,31)(29,34)(30,33)(35,48)(36,40)(38,41)(43,46)(45,47);(3,10)(4,14)(6,15)(7,13)(9,16)(11,12)(19,28)(23,39)(27,45)(29,38)(30,37)(32,48)(40,44)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399060);