#501366

order	512
nilpotency class	4
isoclinism	isoclinic class of #501366
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24531
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,15,19,9)(4,20,22,44)(5,12)(6,27)(7,14)(8,11)(10,23,38,32)(13,31)(16,47)(17,39,25,34)(18,35,52,46)(21,53)(24,42,57,54)(26,43)(28,41,59,60)(29,61,58,37)(30,40)(33,49)(36,56)(45,50)(48,62)(51,64)(55,63);(1,3)(2,9)(4,21)(5,17)(6,28)(7,25)(8,19)(10,36)(11,15)(12,34)(13,41)(14,39)(16,24)(18,30)(20,53)(22,55)(23,51)(26,52)(27,60)(29,50)(31,59)(32,56)(33,37)(35,43)(38,64)(40,46)(42,47)(44,63)(45,61)(48,57)(49,58)(54,62);(1,4,7,32,8,22,5,23)(2,10,12,20,11,38,14,44)(3,16,17,49,19,48,25,50)(6,29,30,57,31,58,26,24)(9,33,39,47,15,45,34,62)(13,42,40,37,27,54,43,61)(18,51,28,55,52,56,59,21)(35,53,60,36,46,63,41,64);(1,3)(2,9)(4,16)(5,25)(6,18)(7,17)(8,19)(10,33)(11,15)(12,39)(13,35)(14,34)(20,47)(21,24)(22,48)(23,50)(26,59)(27,46)(28,30)(29,51)(31,52)(32,49)(36,37)(38,45)(40,60)(41,43)(42,53)(44,62)(54,63)(55,57)(56,58)(61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408562);