#137703

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1898
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9079
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44899);