#140691

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁴	#131
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,5,7,8)(4,6)(9,10,13,19,15,20,31,44,14,18,21,41,32,42,46,69)(11,16,24,35,25,36,52,64,23,34,30,43,51,63,60,71)(12,17,28,38,29,40,58,61,27,39,45,65,56,68,48,72)(22,47,50,70,26,53,55,67,49,33,59,62,54,37,57,66);(1,3,4,7)(2,5)(6,8)(9,11,21,24,15,25,46,52,14,23,13,30,32,51,31,60)(10,16,19,43,20,36,44,71,18,34,41,35,42,63,69,64)(12,26,28,57,29,49,58,50,27,54,45,55,56,22,48,59)(17,37,65,66,40,47,72,70,39,53,38,67,68,33,61,62);(2,6)(5,8)(9,12,14,27)(10,17,18,39)(11,22)(13,28,21,45)(15,29,32,56)(16,33)(19,38,41,65)(20,40,42,68)(23,49)(24,50)(25,26)(30,59)(31,58,46,48)(34,47)(35,62)(36,37)(43,70)(44,61,69,72)(51,54)(52,55)(53,63)(57,60)(64,66)(67,71)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg,e^b=ei,f^b=fg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47887);