#148639

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55835);