#153501

order	512
nilpotency class	6
isoclinism	isoclinic class of #153493
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13569
commutator	#125	C₂²×C₄
Frattini, cosocle	#491	C₂³
prime radical	#532	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3142 ◁ G
LECS	G ▷ #491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,d^c=di,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60697);