#113874

order	512
nilpotency class	3
isoclinism	isoclinic class of #113863
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21070);