#144074

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

Polycyclic presentation:

2^9|a^2=ghi,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, 51270);