#145453

order	512
nilpotency class	4
isoclinism	isoclinic class of #145363
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

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, 52649);