#135528

order	512
nilpotency class	4
isoclinism	isoclinic class of #135527
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8747
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

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

normal subgroups only

This group has 451 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)(4,18)(5,22,25,10)(6,11)(7,12)(8,15,31,47)(9,20)(14,41)(16,39)(17,40)(19,52,23,50)(21,51)(24,55,54,35)(26,58)(27,33)(28,37)(29,59,62,64)(30,46,36,57)(32,53,34,56)(38,43)(42,61,45,49)(44,63)(48,60);(1,3,7,17)(2,8,12,31)(4,14,21,44)(5,15,25,47)(6,26,28,16)(9,29,33,62)(10,13,22,40)(11,36,37,30)(18,49,51,61)(19,42,23,45)(20,48,27,43)(24,57,54,46)(32,60,34,38)(35,58,55,39)(41,52,63,50)(53,59,56,64);(1,4,5,23)(2,9,10,34)(3,14,15,45)(6,27,24,53)(7,21,25,19)(8,29,13,38)(11,18,35,52)(12,33,22,32)(16,48,46,64)(17,44,47,42)(20,54,56,28)(26,43,57,59)(30,61,39,41)(31,62,40,60)(36,49,58,63)(37,51,55,50)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,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, 42724);