#150128

order	512
nilpotency class	5
isoclinism	isoclinic class of #150123
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	#2215	C₄
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

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

normal subgroups only

This group has 202 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)(19,30,33,61)(20,26)(21,38)(22,43)(23,47)(24,29)(25,32,49,66)(27,40)(28,45)(31,59,35,71)(34,70,69,80)(36,72)(37,57)(39,56,67,60)(41,78)(42,58)(44,62,64,53)(46,54)(48,63,51,68)(50,76,79,73)(52,74)(55,65)(75,77);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,25)(20,35)(21,39)(22,44)(23,31)(24,33)(26,51)(27,53)(28,56)(29,49)(30,58)(32,65)(34,45)(36,73)(37,69)(38,61)(40,66)(41,76)(42,67)(43,50)(46,64)(47,48)(52,70)(54,79)(55,62)(57,60)(59,72)(63,74)(68,77)(71,78)(75,80);(3,9)(6,12)(10,15)(13,16)(17,20,24,23)(18,26,29,47)(19,31,33,35)(21,36,42,41)(22,37,46,45)(25,48,49,51)(27,52,55,75)(28,43,57,54)(30,59,61,71)(32,63,66,68)(34,64,69,44)(38,72,58,78)(39,76,67,73)(40,74,65,77)(50,56,79,60)(53,80,62,70)

Polycyclic presentation:

2^9|a^2=e,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,d^b=dfi,e^b=ei,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234337, #72234342, #72234347, …
maximal normal subgroup of	#72234337, #72234342, #72234347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57324);