#108321

order	512
nilpotency class	3
isoclinism	isoclinic class of #108316
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

696 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,e^b=ei,f^b=fhi,d^c=di,e^c=ei,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, 15517);