#501369

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	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 636 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)(3,9,17,40)(4,21,27,28)(5,11,10,25)(6,14)(7,12)(8,32)(13,30)(15,47)(16,36)(18,37,20,39)(19,49,53,38)(22,52)(23,24)(26,61,48,46)(29,34,42,43)(31,60)(33,63,55,45)(35,57)(41,58)(44,56)(50,62)(51,64)(54,59);(1,3,6,17)(2,9,14,40)(4,22,27,59)(5,16,10,41)(7,18,23,20)(8,33,30,55)(11,36,25,58)(12,37,24,39)(13,45,32,63)(15,48,51,26)(19,56,53,31)(21,54,28,52)(29,62,42,35)(34,50,43,57)(38,60,49,44)(46,47,61,64);(1,4,23,11,6,27,7,25)(2,10,12,21,14,5,24,28)(3,15,18,50,17,51,20,57)(8,34,56,26,30,43,31,48)(9,35,39,64,40,62,37,47)(13,46,44,29,32,61,60,42)(16,49,54,63,41,38,52,45)(19,36,55,22,53,58,33,59);(1,5)(2,11)(3,16)(4,24)(6,10)(7,21)(8,29)(9,36)(12,27)(13,43)(14,25)(15,49)(17,41)(18,54)(19,47)(20,52)(22,39)(23,28)(26,60)(30,42)(31,46)(32,34)(33,62)(35,55)(37,59)(38,51)(40,58)(44,48)(45,57)(50,63)(53,64)(56,61)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,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, 408565);