#113772

order	512
nilpotency class	3
isoclinism	isoclinic class of #113765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20968);