#10220391

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	#48823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10127587);