#150793

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
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

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

normal subgroups only

This group has 212 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)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,21,28,24,32,49,31)(19,33,36,69,39,71,47,70)(20,40,43,76,46,63,80,77)(22,50,48,29,23,52,42,30)(25,54,55,35,59,73,58,34)(26,60,62,44,27,65,64,45)(37,74,51,61,38,75,53,66)(41,67,78,56,72,68,79,57);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,25)(20,41)(21,47)(22,51)(23,53)(24,39)(26,61)(27,66)(28,58)(29,67)(30,68)(31,55)(32,59)(33,63)(34,44)(35,45)(36,49)(37,48)(38,42)(40,71)(43,79)(46,72)(50,56)(52,57)(54,60)(62,74)(64,75)(65,73)(69,76)(70,77)(78,80);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,42,60,21,43,22,44,24,46,48,65,49,80,23,45)(18,26,52,77,31,64,29,63,32,27,50,76,28,62,30,40)(19,34,53,79,36,54,37,41,39,35,51,78,47,73,38,72)(25,33,68,75,58,70,56,61,59,71,67,74,55,69,57,66)

Polycyclic presentation:

2^9|a^2=e,c^2=fhi,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=cf,d^b=dg,e^b=efgh,f^b=fh,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, 57989);