#148490

order	512
nilpotency class	4
isoclinism	isoclinic class of #148489
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11174
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 752 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)(3,14,19,38)(4,20)(5,26,7,11)(6,28,31,12)(8,13)(9,16)(10,23,37,56)(15,43,46,50)(17,48)(18,34)(21,52)(22,51,24,57)(25,40)(27,39)(29,42)(30,41)(32,61,63,53)(33,47,59,49)(35,60)(36,54)(44,62)(45,64)(55,58);(1,3,8,19)(2,9,13,34)(4,21,25,44)(5,18,7,16)(6,29,31,17)(10,35,37,64)(11,38,26,14)(12,33,28,59)(15,22,46,24)(20,50,40,43)(23,53,56,61)(27,47,30,49)(32,58,63,36)(39,48,41,42)(45,54,60,55)(51,62,57,52);(1,4,5,22,8,25,7,24)(2,10,11,36,13,37,26,58)(3,15,16,44,19,46,18,21)(6,23,30,55,31,56,27,54)(9,32,14,35,34,63,38,64)(12,40,41,57,28,20,39,51)(17,45,49,53,29,60,47,61)(33,52,48,43,59,62,42,50)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55686);