#150119

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#2215	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=df,f^b=fh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57315);