#10218653

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	#56018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 10125849);