#149319

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

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

normal subgroups only

This group has 603 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)(3,15,20,45)(4,21)(5,27,30,11)(6,31,29,12)(7,33)(8,14)(9,17)(10,24,44,60)(13,36)(16,50,53,41)(18,47)(19,38,34,46)(22,56,54,37)(23,48,58,43)(25,61)(26,55)(28,40)(32,39)(35,42)(49,57)(51,62)(52,63)(59,64);(1,3,7,34,8,20,36,19)(2,9,33,39,14,40,13,47)(4,22,35,53,26,54,25,16)(5,28,29,18,30,17,6,32)(10,41,43,37,44,50,48,56)(11,45,31,38,27,15,12,46)(21,49,42,64,55,63,61,62)(23,52,60,51,58,57,24,59);(1,4,7,35,8,26,36,25)(2,10,33,43,14,44,13,48)(3,16,34,22,20,53,19,54)(5,23,29,60,30,58,6,24)(9,37,39,50,40,56,47,41)(11,42,31,55,27,61,12,21)(15,49,46,64,45,63,38,62)(17,51,32,57,28,59,18,52)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,f^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, 56515);