#479818

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17263
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 144 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)(4,21)(5,27,30,41)(6,13)(7,12)(8,14)(9,18)(10,24)(11,29,42,31)(16,36)(17,48,50,40)(19,34)(20,35)(22,32)(23,37,55,63)(25,39)(26,38)(28,44,51,33)(43,47)(45,61,49,59)(46,60)(52,54,57,56)(53,62,58,64);(1,3)(2,9,14,15)(4,22)(5,28,30,51)(6,20)(7,19)(8,18)(10,36,39,60)(11,40)(12,35,13,34)(16,25)(17,29,50,31)(21,43,38,32)(23,49,55,45)(24,46)(26,47)(27,33)(37,61)(41,44)(42,48)(52,64)(53,54,58,56)(57,62)(59,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,39,13,38)(3,16,19,47,18,46,20,22)(5,23,29,54,30,55,31,56)(9,32,34,60,15,43,35,36)(11,37,41,52,42,63,27,57)(17,49,51,53,50,45,28,58)(33,61,48,62,44,59,40,64);(1,5)(2,11)(3,17,18,50)(4,23)(6,29)(7,31)(8,30)(9,33,15,44)(10,37)(12,41)(13,27)(14,42)(16,45,46,49)(19,51,20,28)(21,52)(22,53,47,58)(24,54)(25,56)(26,55)(32,59,43,61)(34,48,35,40)(36,62,60,64)(38,57)(39,63)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,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, 387014);