#506393

order	512
nilpotency class	4
isoclinism	isoclinic class of #501995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28227
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 591 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)(3,11)(4,21)(5,28)(6,32)(7,14)(8,15)(9,18)(10,25)(12,29)(13,33)(16,45)(17,49)(19,43)(20,44)(22,38)(23,57)(24,39)(26,41)(27,40)(30,36)(31,35)(34,56)(37,58)(42,54)(46,48)(47,62)(50,55)(51,60)(52,64)(53,63)(59,61);(1,3,16,30,33,54,29,5,8,20,6,18,7,19,48,31)(2,9,12,11,14,28,45,43,15,36,46,44,13,35,32,42)(4,22,24,56,25,53,59,58,27,52,47,23,26,17,50,51)(10,37,62,49,21,34,61,64,41,60,39,63,40,57,55,38);(1,4)(2,10)(3,17)(5,23)(6,24)(7,26)(8,27)(9,34)(11,38)(12,39)(13,40)(14,21)(15,41)(16,47)(18,51)(19,52)(20,53)(22,54)(25,33)(28,60)(29,50)(30,58)(31,56)(32,61)(35,37)(36,57)(42,63)(43,49)(44,64)(45,55)(46,62)(48,59);(1,5,29,19,7,30,16,20,8,31,48,54,33,18,6,3)(2,11,32,9,13,42,46,35,15,44,45,36,14,43,12,28)(4,23,50,52,26,58,47,53,27,56,59,22,25,51,24,17)(10,38,61,34,40,63,62,37,41,64,55,57,21,49,39,60)

Polycyclic presentation:

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

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, 413589);