#122124

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=i,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, 29320);