#107422

order	512
nilpotency class	3
isoclinism	isoclinic class of #107406
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,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=ei,f^b=fi,d^c=di,e^c=eh

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, 14618);