#501371

order	512
nilpotency class	4
isoclinism	isoclinic class of #501370
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	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 648 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=eh,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,f^c=fi,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, 408567);