#491497

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

514 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,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,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398693);