#136790

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#3150	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 392 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43986);