#148214

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:

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

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,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh,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, 55410);