#148217

order	512
nilpotency class	4
isoclinism	isoclinic class of #140874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11076
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

673 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=ghi,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, 55413);