#508491

order	512
nilpotency class	4
isoclinism	isoclinic class of #508490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29890
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

585 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=ghi,h^c=hi,g^d=gi

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, 415687);