#145467

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:

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

Polycyclic presentation:

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

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, 52663);