#145079

order	512
nilpotency class	4
isoclinism	isoclinic class of #145026
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,5,3)(4,6,8,7)(9,10,15,19)(11,21,23,16)(12,26,28,61)(13,30,35,64)(14,36,33,20)(17,41,44,27)(18,45,48,34)(22,51,52,60)(24,56,31,65)(25,40,55,50)(29,63,37,47)(32,49,68,43)(38,53,42,62)(39,58,46,67)(54,70,59,72)(57,69,66,71);(1,3,5,2)(4,7,8,6)(9,11,14,25,15,23,33,55)(10,16,20,40,19,21,36,50)(12,22,29,54,28,52,37,59)(13,31,34,67,35,24,45,58)(17,42,43,71,44,38,49,69)(18,46,30,56,48,39,64,65)(26,60,47,70,61,51,63,72)(27,62,32,66,41,53,68,57);(2,6)(3,7)(4,8)(9,12)(10,17)(11,22,23,52)(13,32)(14,37)(15,28)(16,38,21,42)(18,47)(19,44)(20,49)(24,57,31,66)(25,59,55,54)(26,30)(27,34)(29,33)(35,68)(36,43)(39,70,46,72)(40,71,50,69)(41,45)(48,63)(51,65,60,56)(53,67,62,58)(61,64)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=d,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52275);