#113858

order	512
nilpotency class	3
isoclinism	isoclinic class of #113854
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

774 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,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=ghi

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, 21054);