#143355

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×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 403 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,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, 50551);