#124769

order	512
nilpotency class	4
isoclinism	isoclinic class of #124650
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1527
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#376	C₂³
socle	C₂	#7618
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,13,16,39,17,34,25,56)(4,9)(5,21,23,54)(6,11)(7,12)(8,15,28,45,30,47,29,46)(10,24,36,49)(14,40,42,62,43,38,51,61)(18,48,22,53)(19,32)(20,33)(26,37)(27,41,57,63,59,44,58,64)(31,50,35,55)(52,60);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,27)(10,34)(11,28)(12,30)(13,36)(16,26)(18,41)(19,51)(20,43)(21,39)(22,44)(23,47)(24,45)(29,37)(31,38)(32,57)(33,59)(35,40)(42,52)(46,49)(48,62)(50,63)(53,61)(54,56)(55,64)(58,60);(1,4,6,19,7,20,26,52)(2,9,11,32,12,33,37,60)(3,14,25,51,17,43,16,42)(5,22,24,55,23,18,49,50)(8,27,28,57,30,59,29,58)(10,35,21,53,36,31,54,48)(13,38,56,62,34,40,39,61)(15,44,45,64,47,41,46,63)

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31965);