#107484

order	512
nilpotency class	3
isoclinism	isoclinic class of #107474
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,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, 14680);