#143375

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
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	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6,10,14)(7,15,18,19)(8,13,9,12)(11,17,20,16)(21,22,26,30)(23,31,33,50)(24,35,37,49)(25,40,38,29)(27,42,43,32)(28,45,47,39)(34,51,41,46)(36,48,52,44);(1,2)(3,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,19,13,15)(21,23,26,33)(22,27,30,43)(24,36,37,52)(25,41,38,34)(28,46,47,51)(29,48,40,44)(31,49,50,35)(32,45,42,39);(2,4)(6,12)(8,9)(11,19)(13,14)(15,20)(16,17)(24,37)(28,47)(31,50)(32,42)(35,49)(36,52)(39,45)(46,51)

Polycyclic presentation:

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

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, 50571);