#9053873

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(9,10,16,20)(11,21,23,14)(12,22,24,17)(13,19,15,18)(25,26,31,29)(27,32,37,41)(28,33,40,42)(30,34,45,43)(35,46,38,48)(36,47,44,50)(39,49,53,54)(51,55,52,56);(3,7)(6,8)(9,11,16,23)(10,14,20,21)(12,19,24,18)(13,17,15,22)(26,29)(32,41)(33,42)(34,43)(46,48)(47,50)(49,54)(55,56);(1,2)(3,6)(4,5)(7,8)(9,12,16,24)(10,17,20,22)(11,18,23,19)(13,21,15,14)(25,27,31,37)(26,32,29,41)(28,38,40,35)(30,44,45,36)(33,48,42,46)(34,50,43,47)(39,51,53,52)(49,55,54,56);(9,13,16,15)(10,18,20,19)(11,17,23,22)(12,21,24,14)(25,28,31,40)(26,33,29,42)(27,35,37,38)(30,39,45,53)(32,46,41,48)(34,49,43,54)(36,51,44,52)(47,55,50,56);(1,3)(2,6)(4,7)(5,8)(9,14)(10,11)(12,13)(15,24)(16,21)(17,18)(19,22)(20,23)(25,29,31,26)(27,32,37,41)(28,33,40,42)(30,43,45,34)(35,48,38,46)(36,47,44,50)(39,49,53,54)(51,56,52,55);(1,2)(3,6)(4,5)(7,8)(9,15,16,13)(10,19,20,18)(11,22,23,17)(12,14,24,21)(25,30,31,45)(26,34,29,43)(27,36,37,44)(28,39,40,53)(32,47,41,50)(33,49,42,54)(35,51,38,52)(46,55,48,56)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=gi,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, 8961069);