#107928

order	512
nilpotency class	3
isoclinism	isoclinic class of #107918
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	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

721 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15124);