#113879

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=h,g^2=h,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, 21075);