#148832

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11222
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,h^c=hi,f^e=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, 56028);