#149320

order	512
nilpotency class	4
isoclinism	isoclinic class of #140675
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11353
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

595 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,f^c=fi,g^c=gh,g^d=gi,g^e=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, 56516);