#492174

order	512
nilpotency class	4
isoclinism	isoclinic class of #491712
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	#493	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 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)(17,18,23,27)(19,24,22,37)(20,30,32,47)(21,33,36,46)(25,40,42,31)(26,29,44,35)(28,45,38,43)(34,39,48,41);(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10)(17,19,23,22)(18,24,27,37)(20,28,32,38)(21,29,36,35)(25,39,42,41)(26,33,44,46)(30,45,47,43)(31,34,40,48);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,20,23,32)(18,25,27,42)(19,28,22,38)(21,34,36,48)(24,39,37,41)(26,43,44,45)(29,40,35,31)(30,46,47,33);(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=gi,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=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

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, 399370);