#148085

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11051
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

653 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 55281);