#150971

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
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

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

normal subgroups only

This group has 197 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58167);