#478252

order	512
nilpotency class	3
isoclinism	isoclinic class of #478251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17137
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 917 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,9,17,31)(4,10)(5,24)(6,14)(7,13)(8,12)(11,25)(15,43,45,29)(16,46,48,58)(18,33,19,32)(20,34,51,59)(21,53)(22,42)(23,36)(26,41)(27,40)(28,37)(30,50,38,49)(35,55)(39,54)(44,61,47,60)(52,63,64,57)(56,62);(1,3)(2,9,12,31)(4,20)(5,16)(6,19)(7,18)(8,17)(10,34,36,59)(11,38,40,30)(13,33,14,32)(15,22)(21,52)(23,51)(24,46,41,58)(25,50)(26,48)(27,49)(28,45)(29,42,43,37)(35,61,39,60)(44,55)(47,54)(53,63,62,57)(56,64);(1,4,8,23)(2,10,12,36)(3,15)(5,21,26,56)(6,22,7,28)(9,29)(11,39,40,35)(13,37,14,42)(16,47)(17,45)(18,20)(19,51)(24,53,41,62)(25,54,27,55)(30,57)(31,43)(32,34)(33,59)(38,63)(44,48)(46,61)(49,52)(50,64)(58,60);(1,5)(2,11,12,40)(3,16)(4,21)(6,27)(7,25)(8,26)(9,30,31,38)(10,35,36,39)(13,41,14,24)(15,44)(17,48)(18,50)(19,49)(20,52)(22,55)(23,56)(28,54)(29,57,43,63)(32,58,33,46)(34,60,59,61)(37,53,42,62)(45,47)(51,64)

Polycyclic presentation:

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

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, 385448);