#107511

order	512
nilpotency class	3
isoclinism	isoclinic class of #107494
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	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 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)(5,11)(6,10)(7,12)(8,15)(9,19)(14,22)(16,26)(18,25)(20,24)(21,23)(27,28)(29,32)(30,31)(33,34,38,43,39,42,37,50)(35,40,46,55,45,56,52,54)(36,47,48,41,49,58,51,57)(44,59,62,63,60,64,61,53);(1,3)(2,8,12,23)(4,14)(5,15)(6,21)(7,16)(9,22,25,32)(10,13,11,26)(17,27,24,31)(18,28)(19,30)(20,29)(33,35)(34,40,42,56)(36,44)(37,46)(38,52)(39,45)(41,53,57,63)(43,54,50,55)(47,59,58,64)(48,61)(49,60)(51,62);(1,4,7,20)(2,9,12,25)(3,14,16,29)(5,19,6,18)(8,22,23,32)(10,17,11,24)(13,27,26,31)(15,30,21,28)(33,36,39,49)(34,41,42,57)(35,44,45,60)(37,51,38,48)(40,53,56,63)(43,58,50,47)(46,62,52,61)(54,64,55,59)

Polycyclic presentation:

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

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, 14707);