#505683

order	512
nilpotency class	4
isoclinism	isoclinic class of #500306
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27645
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,d^2=e,e^2=i,f^2=h,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=fhi,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, 412879);