#152268

order	512
nilpotency class	5
isoclinism	isoclinic class of #149800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2129
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12800
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,15)(4,20,24,54)(5,27)(6,11)(7,13,31,43)(8,14)(9,17)(10,23,37,58)(12,30,39,28)(16,47,49,61)(18,33,52,40)(19,46)(21,55,50,32)(22,42,57,38)(25,36,26,41)(29,35)(34,59,45,51)(44,48,60,63)(53,64,62,56);(1,3,5,17,8,19,6,29)(2,9,11,15,14,35,27,46)(4,21,22,56,26,16,23,48)(7,18,28,51,31,52,30,59)(10,32,36,44,38,61,54,62)(12,40,13,34,39,33,43,45)(20,53,37,55,41,60,42,47)(24,50,57,64,25,49,58,63);(1,4,8,26)(2,10)(3,16,19,21)(5,22,6,23)(7,24,31,25)(9,32)(11,36)(12,41)(13,42)(14,38)(15,44)(17,48,29,56)(18,49,52,50)(20,39)(27,54)(28,57,30,58)(33,53)(34,47)(35,61)(37,43)(40,60)(45,55)(46,62)(51,63,59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

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, 59464);