#150794

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

210 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,21,26,23,29,43,28)(19,30,33,62,36,48,35,69)(20,37,40,70,42,72,76,71)(22,44,39,27,46,57,75,59)(24,47,49,31,51,65,60,32)(25,52,54,63,56,79,55,41)(34,67,64,77,68,80,45,53)(38,58,73,78,66,61,74,50);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,24)(20,38)(21,33)(22,45)(23,36)(25,53)(26,49)(27,58)(28,60)(29,51)(30,37)(31,63)(32,41)(34,39)(35,43)(40,73)(42,66)(44,50)(46,64)(47,52)(48,72)(54,67)(55,80)(56,77)(57,78)(59,61)(62,70)(65,79)(68,75)(69,71)(74,76);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,39,52,21,40,46,63,23,42,75,79,43,76,22,41)(18,25,44,71,28,55,59,72,29,56,57,70,26,54,27,37)(19,31,45,73,35,47,68,38,36,32,64,74,33,65,34,66)(24,48,58,80,49,69,78,53,51,30,61,67,60,62,50,77)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57990);