#491625

order	512
nilpotency class	4
isoclinism	isoclinic class of #491565
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

548 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)(19,28,29,42)(20,25,22,37)(21,35,36,48)(24,38,39,32)(26,34,41,30)(31,43,33,47)(40,45,44,46);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(19,29)(24,39)(28,42)(31,33)(32,38)(40,44)(43,47)(45,46);(2,7)(6,14)(8,12)(9,15)(17,19,30,45,22,33,36,38,23,29,34,46,20,31,21,32)(18,24,35,43,25,40,41,42,27,39,48,47,37,44,26,28);(3,10)(6,14)(9,15)(11,16)(17,20,23,22)(18,25,27,37)(19,31,29,33)(21,34,36,30)(24,40,39,44)(26,35,41,48)(28,43,42,47)(32,46,38,45)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fgh,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,c^b=ci,d^b=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398821);