#138167

order	512
nilpotency class	4
isoclinism	isoclinic class of #138165
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9133
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

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, 45363);