#492204

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	#3391	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^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=dei,e^b=eg,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, 399400);