#106206

order	512
nilpotency class	3
isoclinism	isoclinic class of #106194
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh,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, 13402);