#499706

order	512
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23837
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 419 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283798, #72283839, #72283942, …
maximal normal subgroup of	#72283798, #72283839, #72283942, …
maximal quotient of	C₂×#499706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406902);