#107595

order	512
nilpotency class	3
isoclinism	isoclinic class of #107590
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=di,f^b=fi,e^c=eh,f^c=fhi

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, 14791);