#150994

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

189 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=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, 58190);