#148216

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

665 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=h,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, 55412);