#149278

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11344
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,f^d=fi

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, 56474);