#138945

order	512
nilpotency class	4
isoclinism	isoclinic class of #138944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9259
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 510 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46141);