#9053864

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	#52418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(6,8)(10,13)(11,17)(16,20)(19,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,46)(36,38)(37,47)(39,48)(40,42)(41,49)(43,50)(44,45)(51,55)(52,56)(53,54);(3,7)(6,8)(9,10,14,13)(11,18,17,15)(12,20,21,16)(19,22,23,24)(26,31)(32,36)(33,40)(34,45)(46,48)(47,50)(49,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,15,13,18)(12,19,21,23)(16,22,20,24)(25,27,30,38)(26,32,31,36)(28,39,42,35)(29,43,44,37)(33,48,40,46)(34,50,45,47)(41,51,54,52)(49,55,53,56);(9,10,14,13)(11,15,17,18)(12,16,21,20)(19,22,23,24)(25,28,30,42)(26,33,31,40)(27,35,38,39)(29,41,44,54)(32,46,36,48)(34,49,45,53)(37,51,43,52)(47,55,50,56);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,13,20)(11,19,17,23)(15,22,18,24)(25,26)(27,36)(28,40)(29,34)(30,31)(32,38)(33,42)(35,46)(37,50)(39,48)(41,53)(43,47)(44,45)(49,54)(51,55)(52,56);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,18,17,15)(12,20,21,16)(19,24,23,22)(25,29,30,44)(26,34,31,45)(27,37,38,43)(28,41,42,54)(32,47,36,50)(33,49,40,53)(35,51,39,52)(46,55,48,56)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961060);