#153303

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13424
prime radical	#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 327 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)(3,15,20,41)(4,21)(5,11)(6,28)(7,12)(8,14)(9,17,36,26)(10,23)(13,32)(16,37,22,55)(18,45,50,35)(19,51,31,34)(24,38)(25,60)(27,42)(29,40)(30,43)(33,52,63,49)(39,61)(44,58)(46,57,64,48)(47,53,56,62)(54,59);(1,3)(2,9)(4,22)(5,26)(6,17)(7,31)(8,20)(10,37)(11,41)(12,35)(13,45)(14,36)(15,28)(16,24)(18,30)(19,32)(21,53)(23,47)(25,49)(27,50)(29,56)(33,39)(34,43)(38,62)(40,55)(42,51)(44,63)(46,54)(48,61)(52,59)(57,58)(60,64);(1,4,5,23,8,24,6,29)(2,10,11,38,14,40,28,21)(3,16,17,47,20,22,26,56)(7,25,27,58,32,59,30,61)(9,33,15,46,36,63,41,64)(12,44,42,54,13,39,43,60)(18,48,19,52,50,57,31,49)(34,62,45,37,51,53,35,55)

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^c=ehi,f^c=fi,g^c=gh,f^e=fh,g^f=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, 60499);