#131671

order	512
nilpotency class	4
isoclinism	isoclinic class of #131671
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	#62433	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

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

normal subgroups only

This group has 593 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38867);