#131411

order	512
nilpotency class	4
isoclinism	isoclinic class of #131334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,4,8,6)(5,7)(9,10,15,22,16,23,40,54,17,24,39,53,14,21,13,20)(11,25,29,47,30,51,27,44,31,58,37,18,28,45,65,46)(12,32,35,52,34,50,38,56,36,55,42,48,41,19,33,49)(26,59,62,70,61,69,60,71,63,72,67,68,66,57,64,43);(1,3)(2,6)(4,7)(5,8)(9,11)(10,18)(12,26)(13,37)(14,30)(15,27)(16,28)(17,31)(19,43)(20,51)(21,46)(22,25)(23,44)(24,47)(29,40)(32,68)(33,67)(34,66)(35,60)(36,63)(38,62)(39,65)(41,61)(42,64)(45,54)(48,72)(49,69)(50,71)(52,59)(53,58)(55,70)(56,57);(1,4,5,6)(2,3,7,8)(9,12)(10,19)(11,26)(13,38)(14,41)(15,42)(16,34)(17,36)(18,43)(20,52)(21,55)(22,56)(23,32)(24,50)(25,57)(27,64)(28,66)(29,67)(30,61)(31,63)(33,40)(35,39)(37,62)(44,68)(45,72)(46,70)(47,71)(48,54)(49,53)(51,59)(58,69)(60,65)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38607);