#131059

order	512
nilpotency class	4
isoclinism	isoclinic class of #130738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38255);