#138946

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	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 505 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,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, 46142);