#148082

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
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)(3,14,18,8,19,33,16,34)(4,20,24,41,25,55,22,54)(6,28,26,11,30,40,59,42)(9,35,38,58,39,29,37,23)(15,45,46,62,48,53,21,56)(17,32,51,60,52,64,49,44)(31,61,36,57,63,47,43,50);(1,3,7,19)(2,8)(4,21,25,46)(5,16,27,18)(6,17,30,52)(9,36)(10,14)(11,32)(12,33)(13,34)(15,24,48,22)(20,53)(23,57,58,50)(26,49,59,51)(28,60)(29,47,35,61)(31,38)(37,63)(39,43)(40,44)(41,62)(42,64)(45,55)(54,56);(1,4)(2,9)(3,15)(5,24)(6,29)(7,25)(8,31)(10,38)(11,41)(12,37)(13,39)(14,43)(16,21)(17,50)(18,46)(19,48)(20,28)(22,27)(23,26)(30,35)(32,45)(33,36)(34,63)(40,55)(42,54)(44,56)(47,49)(51,61)(52,57)(53,64)(58,59)(60,62)

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,f^a=fi,h^a=hi,c^b=cf,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, 55278);