#135556

order	512
nilpotency class	4
isoclinism	isoclinic class of #135556
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8749
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,13,17,40)(4,18)(5,22,25,10)(6,11)(7,12)(8,15)(9,20)(14,41,44,63)(16,39,26,58)(19,52,23,50)(21,51)(24,55,54,35)(27,33)(28,37)(29,59)(30,46)(31,47)(32,53,34,56)(36,57)(38,48,60,43)(42,49)(45,61)(62,64);(1,3)(2,8,12,31)(4,14)(5,15)(6,26)(7,17)(9,29,33,62)(10,13,22,40)(11,36,37,30)(16,28)(18,49,51,61)(19,42)(20,48)(21,44)(23,45)(24,57)(25,47)(27,43)(32,60,34,38)(35,58,55,39)(41,52,63,50)(46,54)(53,59)(56,64);(1,4,5,23,7,21,25,19)(2,9,10,34,12,33,22,32)(3,14,15,45,17,44,47,42)(6,27,24,53,28,20,54,56)(8,29,13,38,31,62,40,60)(11,18,35,52,37,51,55,50)(16,48,46,64,26,43,57,59)(30,61,39,41,36,49,58,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fgi,d^c=di,e^c=egi,f^c=fhi,h^c=hi,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42752);