#491555

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	#3169	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

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

normal subgroups only

This group has 520 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,23,27)(19,28,32,45)(20,25,22,37)(21,35,36,48)(24,38,41,31)(26,34,42,29)(30,43,33,44)(39,47,40,46);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16);(3,11)(7,15)(10,16)(12,13)(17,19,29,46,22,30,36,38,23,32,34,47,20,33,21,31)(18,24,35,44,25,40,42,45,27,41,48,43,37,39,26,28);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,23,22)(18,25,27,37)(19,30,32,33)(21,34,36,29)(24,39,41,40)(26,35,42,48)(28,43,45,44)(31,46,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=gi,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, 398751);