#143370

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 50566);