#144158

order	512
nilpotency class	4
isoclinism	isoclinic class of #144123
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51354);