#153093

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 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 373 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:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,e^c=ei

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, 60289);