#491536

order	512
nilpotency class	4
isoclinism	isoclinic class of #491517
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 166 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)(20,26)(21,24)(22,36)(23,37)(25,32)(27,29)(30,43)(31,42)(33,44)(34,39)(35,48)(38,46)(40,45)(41,47);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,24)(25,38)(26,37)(27,39)(28,42)(36,48)(40,41)(43,44);(3,10)(6,15)(9,16)(11,14)(17,19,29,45,23,30,35,46,21,31,34,47,20,33,22,32)(18,25,36,44,26,41,39,42,24,38,48,43,37,40,27,28);(17,20,21,23)(18,26,24,37)(19,30,31,33)(22,34,35,29)(25,40,38,41)(27,36,39,48)(28,43,42,44)(32,45,46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fh,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=cg,d^c=di

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, 398732);