#153272

order	512
nilpotency class	5
isoclinism	isoclinic class of #153086
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33720	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|c^2=d,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,e^b=eg,f^b=fgi,g^b=gh,h^b=hi,e^c=eg,f^c=fgi,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, 60468);