#508319

order	512
nilpotency class	4
isoclinism	isoclinic class of #508316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29732
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32951	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

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, 415515);