#391510

order	512
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2840
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14635
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 775 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,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, 298706);