#131661

order	512
nilpotency class	4
isoclinism	isoclinic class of #131660
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62432	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

578 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fgi

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, 38857);