#10219810

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	#57481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,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, 10127006);