#476492

order	512
nilpotency class	3
isoclinism	isoclinic class of #476491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17091
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1267 conjugacy classes of subgroups — too many to draw. Only its 176 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383688);