#486313

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

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, 393509);