#153114

order	512
nilpotency class	5
isoclinism	isoclinic class of #153086
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2263	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

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

normal subgroups only

This group has 371 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gh,h^b=hi,d^c=dgi,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, 60310);