#107517

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

738 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=hi,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, 14713);