#144060

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
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

341 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 78 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)(5,6,13,20)(7,21)(8,28,34,55)(9,35,11,17)(10,16,39,18)(12,41,38,19)(14,24)(15,37,52,31)(22,43)(23,47)(25,53)(26,54)(27,44)(29,57)(30,51,32,56)(33,50,42,49)(36,46)(40,45)(48,62)(58,68)(59,64)(60,66)(61,67)(63,65);(2,4)(5,7)(6,14)(8,29)(9,36)(10,24)(11,40)(12,26)(13,27)(15,48)(16,53)(17,21)(18,54)(19,46)(20,47)(22,32)(23,39)(25,38)(28,64)(30,61)(31,59)(33,63)(34,58)(35,44)(37,60)(41,45)(42,62)(43,50)(49,67)(51,68)(52,65)(55,66)(56,57);(5,8,13,34)(6,15,20,52)(7,22,27,61)(9,37,11,31)(10,32,39,30)(12,42,38,33)(14,43,47,67)(16,28,18,55)(17,49,35,50)(19,56,41,51)(21,57,44,68)(23,62,24,63)(25,60,26,59)(29,36,58,40)(45,64,46,66)(48,53,65,54)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51256);