#491615

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

552 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,23,27)(19,28,33,45)(20,25,22,37)(21,34,36,39)(24,38,41,31)(26,29,42,35)(30,43,32,48)(40,47,44,46);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15);(2,7)(6,14)(8,12)(9,15)(17,19,29,46,22,32,21,31,23,33,35,47,20,30,36,38)(18,24,39,48,25,40,26,28,27,41,34,43,37,44,42,45);(3,10)(6,14)(9,15)(11,16)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,29,36,35)(24,40,41,44)(26,34,42,39)(28,43,45,48)(31,46,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fghi,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=dg

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, 398811);