#144064

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
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

380 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 98 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)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22,26,30)(23,31,34,50)(24,36,39,49)(25,40,38,29)(27,42,44,33)(28,46,47,37)(32,48,51,45)(35,52,41,43);(2,4)(21,23,26,34)(22,27,30,44)(24,32,39,51)(25,41,38,35)(28,43,47,52)(29,48,40,45)(31,49,50,36)(33,46,42,37);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(23,32)(24,37)(25,38)(27,43)(28,36)(29,40)(31,45)(33,35)(34,51)(39,46)(41,42)(44,52)(47,49)(48,50)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51260);