#153328

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

212 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,8,13)(7,19)(9,27,25,60)(10,18,12,16)(11,33,26,45)(14,39)(15,43,37,31)(17,29,38,35)(20,49)(21,36)(22,58)(23,52)(24,53)(28,51)(30,62,32,50)(34,44)(40,57)(41,54)(42,63)(46,56,47,65)(48,61)(55,67)(59,68)(64,66);(1,2)(3,4)(6,13)(7,20,24,23)(14,40,36,41)(15,37)(16,18)(17,38)(19,49,53,52)(21,54,39,57)(22,55,59,34)(27,60)(28,63,61,64)(33,45)(42,48,66,51)(44,58,67,68)(46,47)(50,62);(2,4)(5,7,10,23,8,24,12,20)(6,14,16,40,13,36,18,41)(9,28,30,64,25,61,32,63)(11,34,29,59,26,55,35,22)(15,44,46,58,37,67,47,68)(17,48,45,66,38,51,33,42)(19,50,49,60,53,62,52,27)(21,56,57,43,39,65,54,31)

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=dh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60524);