#153113

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
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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)(7,20)(8,13)(9,30)(10,34)(11,37)(12,19)(14,22)(15,25)(16,31)(17,35)(18,27)(21,43)(23,59)(24,40)(26,62)(28,50)(29,52)(32,54)(33,41)(36,55)(38,51)(39,48)(42,61)(44,63)(45,57)(46,58)(47,66)(49,65)(53,56)(60,67)(64,68);(1,2)(3,4)(6,13)(7,21)(8,25)(9,31)(10,35)(11,27)(14,43)(15,18)(16,50)(17,52)(19,30)(20,53)(22,57)(23,60)(24,36)(26,63)(28,39)(29,65)(32,56)(33,61)(34,41)(37,42)(40,44)(45,51)(46,55)(47,68)(48,49)(54,67)(58,64)(59,62);(2,4)(5,7)(6,14)(8,26)(9,32)(10,36)(11,38)(12,24)(13,40)(15,47)(16,51)(17,53)(18,54)(19,46)(20,30)(21,35)(22,31)(23,27)(25,58)(28,60)(29,57)(33,64)(34,44)(37,45)(39,66)(41,55)(42,67)(43,52)(48,68)(49,59)(50,62)(56,61)(63,65)

Polycyclic presentation:

2^9|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, 60309);