#506296

order	512
nilpotency class	4
isoclinism	isoclinic class of #499952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28156
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 142 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)(5,13)(6,28)(7,30)(8,14)(9,18)(10,22)(11,26)(12,29)(16,47)(17,53)(19,42)(20,46)(23,43)(24,58)(25,38)(27,44)(31,35)(32,39)(33,61)(34,54)(36,55)(37,56)(40,60)(41,59)(45,50)(48,62)(49,57)(51,64)(52,63);(1,3)(2,9,14,36)(4,16)(5,17)(6,18)(7,31)(8,20)(10,37,40,33)(11,34,30,53)(12,15,28,46)(13,42,44,35)(19,26)(21,57,38,64)(22,48)(23,49)(24,51)(25,52)(27,54)(29,55)(32,61)(39,62,41,45)(43,63,58,47)(50,60)(56,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,32)(8,25)(9,33)(11,38)(12,39)(13,43)(14,40)(15,45)(17,48)(18,49)(19,56)(20,52)(21,30)(24,29)(26,59)(27,60)(28,41)(31,61)(34,57)(35,63)(36,37)(42,47)(44,58)(46,62)(50,54)(51,55)(53,64);(1,5,6,26,8,27,29,7)(2,11,12,13,14,30,28,44)(3,17,18,19,20,54,55,31)(4,22,23,59,25,60,24,32)(9,34,15,42,36,53,46,35)(10,38,39,43,40,21,41,58)(16,48,49,56,52,50,51,61)(33,57,45,47,37,64,62,63)

Polycyclic presentation:

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

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, 413492);