#142914

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
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	C₂³×C₈	C₂³
prime radical	#30253	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 483 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) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fg,g^c=gi,h^c=hi

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, 50110);