#144055

order	512
nilpotency class	4
isoclinism	isoclinic class of #144055
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10267
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^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=fg,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, 51251);