#136876

order	512
nilpotency class	4
isoclinism	isoclinic class of #136874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8977
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,8,14,31)(4,17,18,38,21,53,20,52)(5,22)(6,10)(7,12)(9,19,34,24,35,56,33,55)(11,25)(13,41,42,50,44,64,49,63)(15,46,45,30)(16,32,28,40)(23,57,59,37)(26,51)(27,36)(29,58,61,48,62,43,60,54)(39,47);(1,3,6,14)(2,8)(4,13,21,44)(5,23,25,59)(7,28,27,16)(9,29)(10,31)(11,37)(12,40)(15,47,45,26)(17,50)(18,42,20,49)(19,54,56,48)(22,57)(24,58,55,43)(30,39)(32,36)(33,60)(34,61)(35,62)(38,64)(41,52)(46,51)(53,63);(1,4)(2,9)(3,13)(5,24)(6,21)(7,18)(8,29)(10,35)(11,38)(12,34)(14,44)(15,48)(16,49)(17,51)(19,47)(20,27)(22,52)(23,58)(25,55)(26,56)(28,42)(30,63)(31,62)(32,60)(33,36)(37,64)(39,53)(40,61)(41,57)(43,59)(45,54)(46,50)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44072);