#492003

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	#19962	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

431 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,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,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, 399199);