#122121

order	512
nilpotency class	3
isoclinism	isoclinic class of #122117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

585 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)(3,6,10,15)(4,7)(5,8)(9,13)(11,16,12,14)(17,18,21,26,22,25,36,40)(19,27,30,41,29,23,34,38)(20,31,33,42,32,24,35,39)(28,43,45,48,44,37,46,47);(1,3,5,10)(2,6)(4,12,13,11)(7,14)(8,15)(9,16)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,27)(26,41)(30,36)(31,43)(32,44)(33,46)(35,45)(38,40)(39,48)(42,47);(2,7,8,9)(3,10)(6,14,15,16)(11,12)(17,20,22,32)(18,24,25,31)(19,28,29,44)(21,35,36,33)(23,37,27,43)(26,42,40,39)(30,46,34,45)(38,48,41,47)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,e^2=i,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,g^b=gh,d^c=dhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29317);