#491914

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	C₂×D₈	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,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, 399110);