#153318

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
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

259 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gh,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fgh,g^b=gh,h^b=hi,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, 60514);