#143371

order	512
nilpotency class	4
isoclinism	isoclinic class of #143209
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	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50567);