#140566

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

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, 47762);