#151323

order	512
nilpotency class	5
isoclinism	isoclinic class of #151322
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12186
commutator, Frattini, cosocle	#408	C₂³
prime radical	#13350	C₂
LCS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,g^b=gi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58519);