#491595

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
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

481 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^2=fhi,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, 398791);