#107551

order	512
nilpotency class	3
isoclinism	isoclinic class of #107542
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	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,11,13,15)(4,7)(5,9)(6,10)(8,12,16,14)(17,18,21,26,22,25,35,40)(19,27,30,43,29,23,36,38)(20,31,32,41,33,24,34,39)(28,37,45,47,44,42,46,48);(2,7,10,9)(8,11,16,15)(17,19)(18,23,25,27)(20,28)(21,30)(22,29)(24,37,31,42)(26,38,40,43)(32,45)(33,44)(34,46)(35,36)(39,47,41,48);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,15)(10,16)(17,20,22,33)(18,24,25,31)(19,28,29,44)(21,34,35,32)(23,37,27,42)(26,41,40,39)(30,46,36,45)(38,48,43,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14747);