#149027

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	#11287
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,13)(4,18)(5,23,25,10)(6,26,28,11)(7,12)(8,15,31,45)(9,21,33,41)(14,40,19,51)(16,39,48,60)(17,34)(20,50,54,61)(22,36)(24,35)(27,37)(29,58,59,64)(30,47)(32,53)(38,43)(42,63)(44,57)(46,62)(49,52)(55,56);(1,3,6,16,7,17,28,48)(2,8,11,30,12,31,26,57)(4,19,21,46,22,14,41,43)(5,15,24,44,25,45,27,47)(9,29,18,49,33,59,36,63)(10,34,35,60,23,13,37,39)(20,42,53,58,54,52,55,64)(32,38,61,40,56,62,50,51);(1,4,5,20,7,22,25,54)(2,9,10,32,12,33,23,56)(3,14,15,42,17,19,45,52)(6,21,27,55,28,41,24,53)(8,29,13,38,31,59,34,62)(11,36,37,61,26,18,35,50)(16,46,47,58,48,43,44,64)(30,49,60,51,57,63,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^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=dhi,f^b=fhi,g^b=ghi,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,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, 56223);