#509847

order	512
nilpotency class	4
isoclinism	isoclinic class of #509118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#31056
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417043);