#113883

order	512
nilpotency class	3
isoclinism	isoclinic class of #113877
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

804 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:

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

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=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21079);