#107450

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

743 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)(3,5)(4,12)(6,9)(7,10)(8,14)(11,15)(13,16)(17,18,21,28,23,26,22,27)(19,29,32,42,33,24,31,39)(20,34,36,40,37,41,35,25)(30,44,45,43,46,48,47,38);(1,3,5,10,7,9,6,2)(4,11,14,16,15,12,13,8)(17,19,22,32,23,33,21,31)(18,24,27,39,26,29,28,42)(20,30,35,45,37,46,36,47)(25,38,41,44,40,43,34,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,40)(27,41)(28,34)(29,43)(31,47)(32,45)(33,46)(39,44)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14646);