#148394

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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)(3,16,21,48)(4,22,27,45)(5,29)(6,13)(7,33)(8,35)(9,15)(10,18,39,54)(11,25,44,28)(12,30)(14,31)(17,49)(19,46,32,38)(20,40,34,37)(23,58)(24,59,60,42)(26,43,61,57)(36,50)(41,51)(47,55)(52,62)(53,63)(56,64);(1,3,5,18,8,19,30,20,9,21,31,54,6,32,7,34)(2,10,12,16,13,40,29,38,15,39,33,48,35,37,14,46)(4,23,24,52,28,56,61,55,27,53,60,17,25,50,26,51)(11,41,42,64,22,49,57,58,44,47,59,36,45,62,43,63);(1,4)(2,11)(3,17)(5,26)(6,28)(7,24)(8,25)(9,27)(10,36)(12,43)(13,45)(14,42)(15,44)(16,47)(18,53)(19,55)(20,56)(21,52)(22,35)(23,54)(29,59)(30,60)(31,61)(32,51)(33,57)(34,50)(37,63)(38,49)(39,64)(40,58)(41,48)(46,62)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,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, 55590);