#143868

order	512
nilpotency class	4
isoclinism	isoclinic class of #143819
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,22,26)(19,27,28,45)(20,31,32,47)(21,36,34,25)(23,38,39,29)(24,41,42,35)(30,46,37,43)(33,44,48,40);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(20,32)(24,42)(27,45)(29,38)(31,47)(33,48)(35,41)(43,46);(4,14)(6,15)(7,12)(10,13)(17,19)(18,23)(20,33)(21,37)(22,28)(24,43)(25,44)(26,39)(27,31)(29,35)(30,34)(32,48)(36,40)(38,41)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=dgi,e^c=eh,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51064);