#145504

order	512
nilpotency class	4
isoclinism	isoclinic class of #145502
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	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,6)(4,7)(5,8)(9,10,18,25)(11,26,32,56)(12,33,40,64)(13,41,42,22)(14,43,15,21)(16,24,17,23)(19,44,50,28)(20,37,55,36)(27,45,61,71)(29,49,62,48)(30,47,31,46)(34,57,66,72)(35,54,67,53)(38,63,39,52)(51,58,68,65)(59,70,60,69);(1,2)(3,7)(4,6)(5,8)(9,11,18,32)(10,19,25,50)(12,34,40,66)(13,29,42,62)(14,44,15,28)(16,31,17,30)(20,51,55,68)(21,46,43,47)(22,56,41,26)(23,49,24,48)(27,36,61,37)(33,45,64,71)(35,59,67,60)(38,58,39,65)(52,70,63,69)(53,72,54,57);(2,6)(3,4)(7,8)(9,12,13,35,17,39,14,36,18,40,42,67,16,38,15,37)(10,20,21,52,24,54,22,33,25,55,43,63,23,53,41,64)(11,27,28,58,31,60,62,66,32,61,44,65,30,59,29,34)(19,45,26,57,49,70,47,51,50,71,56,72,48,69,46,68)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=hi,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=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh

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, 52700);