#501373

order	512
nilpotency class	4
isoclinism	isoclinic class of #501370
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24531
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

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, 408569);