#151069

order	512
nilpotency class	5
isoclinism	isoclinic class of #151066
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#367
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#367
LCS	G ▷ C₂×C₁₆ ▷ 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

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

normal subgroups only

This group has 127 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=dfh,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgi,e^b=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234337, #72234344, #72234349, …
maximal normal subgroup of	#72234337, #72234344, #72234349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58265);