#145390

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

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

normal subgroups only

This group has 314 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)(7,8)(9,10)(11,26)(12,34)(13,42)(14,45)(15,22)(16,24)(17,23)(18,25)(19,28)(20,37)(21,44)(27,47)(29,49)(30,51)(31,48)(32,57)(33,58)(35,68)(36,54)(38,55)(39,53)(40,67)(41,59)(43,50)(46,56)(52,61)(60,69)(62,66)(63,70)(64,71)(65,72);(2,5)(6,7)(9,11,18,33)(10,19,25,51)(12,35,41,60)(13,43,15,29)(14,30,44,28)(16,32,17,31)(20,52,56,72)(21,57,45,48)(22,58,42,26)(23,50,24,49)(27,36,64,38)(34,66,59,70)(37,62,46,63)(39,61,40,65)(47,67,71,53)(54,68,55,69);(2,6)(4,8)(5,7)(9,12,14,46,16,39,13,36,18,41,44,37,17,40,15,38)(10,20,22,59,23,54,21,53,25,56,42,34,24,55,45,67)(11,27,29,65,31,62,28,60,33,64,43,61,32,63,30,35)(19,47,48,68,49,70,26,52,51,71,57,69,50,66,58,72)

Polycyclic presentation:

2^9|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, 52586);