#149877

order	512
nilpotency class	5
isoclinism	isoclinic class of #149799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11534
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2961 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dh,e^b=efhi,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57073);