#144073

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51269);