#136864

order	512
nilpotency class	4
isoclinism	isoclinic class of #136863
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8973
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

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

normal subgroups only

This group has 358 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)(4,17,20,37,21,50,18,51)(5,22)(6,10)(7,12)(9,19,33,24,34,53,40,52)(11,26)(13,41,44,62,45,63,42,49)(15,47,46,30)(16,32,28,39)(23,54,56,36)(25,35)(27,55)(29,58,59,48,61,43,60,64)(38,57);(1,3)(2,8,10,31)(4,13)(5,23)(6,14)(7,28)(9,29,34,61)(11,36,22,54)(12,39,35,32)(15,27)(16,25)(17,49,50,62)(18,42)(19,48)(20,44)(21,45)(24,43)(26,56)(30,55,47,38)(33,59,40,60)(37,41,51,63)(46,57)(52,58)(53,64);(1,4)(2,9)(3,13)(5,24)(6,21)(7,18)(8,29)(10,34)(11,37)(12,40)(14,45)(15,48)(16,44)(17,38)(19,27)(20,25)(22,51)(23,43)(26,52)(28,42)(30,49)(31,61)(32,59)(33,35)(36,41)(39,60)(46,64)(47,62)(50,55)(53,57)(54,63)(56,58)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gi,e^2=hi,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=eh,f^c=fhi,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, 44060);