#144447

order	512
nilpotency class	4
isoclinism	isoclinic class of #142232
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,e^c=eg,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51643);