#491945

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

422 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)(17,18,21,24)(19,28,30,45)(20,26,23,37)(22,34,33,47)(25,41,39,31)(27,32,40,36)(29,46,38,44)(35,42,48,43);(1,2)(3,7)(4,8)(5,9)(6,12)(10,14)(11,15)(13,16)(18,24)(25,39)(26,37)(27,40)(28,45)(34,47)(42,43)(44,46);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,19)(18,25)(20,29)(21,30)(22,35)(23,38)(24,39)(26,42)(27,44)(28,34)(31,36)(32,41)(33,48)(37,43)(40,46)(45,47);(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)(25,42,39,43)(27,34,40,47)(28,46,45,44)(31,35,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=g,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, 399141);