#479577

order	512
nilpotency class	3
isoclinism	isoclinic class of #479233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2785
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17216
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

841 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 841 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,13)(4,18)(5,22)(6,27)(7,11)(8,16)(9,20)(10,26)(12,29)(14,30)(15,44)(17,32)(19,49)(21,35)(23,54)(24,55)(25,39)(28,38)(31,47)(33,42)(34,53)(36,45)(37,51)(40,46)(41,60)(43,59)(48,52)(50,64)(56,61)(57,62)(58,63);(1,3)(2,8,11,13)(4,14)(5,23,25,15)(6,17)(7,16)(9,33,18,30)(10,36)(12,32,27,40)(19,50,24,41)(20,42)(21,43)(22,44)(26,47,28,45)(29,46)(31,38)(34,56)(35,59,48,57)(37,58)(39,54)(49,64)(51,63,53,61)(52,62)(55,60);(1,4)(2,9,11,18)(3,14)(5,24,25,19)(6,21)(7,20)(8,30,13,33)(10,37)(12,35,27,48)(15,41,23,50)(16,42)(17,43)(22,49)(26,53,28,51)(29,52)(31,56)(32,57,40,59)(34,38)(36,58)(39,55)(44,64)(45,61,47,63)(46,62)(54,60);(1,5)(2,10)(3,15)(4,19)(6,28)(7,25)(8,31)(9,34)(11,38)(12,39)(13,36)(14,41)(16,23)(17,47)(18,37)(20,24)(21,53)(22,27)(26,29)(30,56)(32,44)(33,58)(35,49)(40,54)(42,50)(43,63)(45,46)(48,55)(51,52)(57,64)(59,60)(61,62)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386773);