#491594

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	#33733	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 515 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)(19,28)(20,25)(21,34)(22,37)(23,27)(24,32)(26,35)(29,42)(30,41)(31,43)(33,48)(36,39)(38,45)(40,46)(44,47);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(19,29)(24,38)(28,42)(31,33)(32,45)(40,44)(43,48)(46,47);(17,19,30,46,22,33,21,32,23,29,35,47,20,31,36,45)(18,24,39,48,25,40,26,28,27,38,34,43,37,44,41,42);(3,10)(6,15)(9,16)(11,14)(17,20,23,22)(18,25,27,37)(19,31,29,33)(21,30,36,35)(24,40,38,44)(26,34,41,39)(28,43,42,48)(32,46,45,47)

Polycyclic presentation:

2^9|a^2=g,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, 398790);