#484557

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

783 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18)(19,31)(20,36)(21,40)(22,30)(23,29)(24,27)(25,33)(26,38)(28,43)(32,59)(34,65)(35,58)(37,68)(39,55)(41,56)(42,52)(44,54)(45,50)(46,53)(47,74)(48,64)(49,72)(51,71)(57,61)(60,66)(62,79)(63,77)(67,70)(69,75)(73,78)(76,80);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,24,35)(18,25,27,48)(20,32,39,63)(21,41,44,72)(22,34,23,46)(26,47,50,75)(28,53,52,65)(29,49,30,56)(31,54,58,40)(33,42,64,43)(36,66,55,76)(37,69,71,74)(38,61,45,62)(51,77,68,59)(57,78,79,67)(60,73,80,70);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,26)(19,32)(21,37)(22,45)(23,38)(24,39)(25,47)(27,50)(28,51)(29,55)(30,36)(31,57)(33,60)(34,62)(35,63)(40,67)(41,69)(42,73)(43,70)(44,71)(46,61)(48,75)(49,76)(52,68)(53,77)(54,78)(56,66)(58,79)(59,65)(64,80)(72,74);(18,27)(25,48)(26,50)(28,52)(29,30)(31,58)(36,55)(40,54)(47,75)(49,56)(51,68)(53,65)(57,79)(59,77)(66,76)(67,78)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391753);