#122123

order	512
nilpotency class	3
isoclinism	isoclinic class of #122122
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

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,g^b=gh,d^c=di,e^c=eh,g^c=ghi

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, 29319);