#513132

order	512
nilpotency class	5
isoclinism	isoclinic class of #513086
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420328);