#388003

order	512
nilpotency class	3
isoclinism	isoclinic class of #388001
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14507
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

743 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295199);