#9434193

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(2,5)(3,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,28)(27,32,35,38)(29,39,42,33)(30,34,43,40)(36,49,47,44)(37,45,48,51)(41,52,53,46)(50,56,55,54);(2,5)(3,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(27,35)(30,43)(32,38)(34,40)(36,47)(41,53)(44,49)(46,52);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,35)(26,32,28,38)(29,36,42,47)(30,37,43,48)(33,44,39,49)(34,45,40,51)(41,50,53,55)(46,54,52,56);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,26)(27,32,35,38)(29,33,42,39)(30,40,43,34)(36,49,47,44)(37,45,48,51)(41,46,53,52)(50,56,55,54);(1,3)(2,6)(4,7)(5,8)(25,29,31,42)(26,33,28,39)(27,36,35,47)(30,41,43,53)(32,44,38,49)(34,46,40,52)(37,50,48,55)(45,54,51,56);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,43)(26,34,28,40)(27,37,35,48)(29,41,42,53)(32,45,38,51)(33,46,39,52)(36,50,47,55)(44,54,49,56)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=ghi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341389);