#153490

order	512
nilpotency class	6
isoclinism	isoclinic class of #153484
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1404
commutator	#125	C₂²×C₄
Frattini, cosocle	#491	C₂³
prime radical	#3553	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#491 ◁ G
LECS	G ▷ #491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

280 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=e,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=ci,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60686);