#10218654

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

4066 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)(11,21)(12,18)(13,23)(14,17)(15,19)(16,20)(22,24)(25,26)(27,29)(28,39)(30,46)(31,32)(33,38)(34,40)(35,48)(36,42)(37,47)(41,44)(43,49)(45,50)(51,56)(52,53)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,16,14)(10,17,20,21)(12,19,13,22)(15,18,24,23)(26,32)(28,40)(29,33)(35,46)(36,41)(43,53)(45,47)(51,54);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,20,23)(11,19,14,22)(15,17,24,21)(25,27)(26,33)(28,41)(29,32)(30,37)(31,38)(34,44)(35,47)(36,40)(39,42)(43,54)(45,46)(48,50)(49,56)(51,53)(52,55);(2,6)(5,7)(9,13,16,12)(10,18,20,23)(11,19,14,22)(15,21,24,17)(25,28)(26,34)(27,36)(29,44)(30,43)(31,40)(32,39)(33,42)(35,49)(37,51)(38,41)(45,56)(46,52)(47,55)(48,53)(50,54);(1,2)(3,5)(4,6)(7,8)(9,14,16,11)(10,17,20,21)(12,22,13,19)(15,18,24,23)(25,29,31,33)(26,27,32,38)(28,42,40,44)(30,47,48,45)(34,41,39,36)(35,50,46,37)(43,56,53,55)(49,51,52,54);(1,3)(2,7)(4,8)(5,6)(9,15,16,24)(10,19,20,22)(11,18,14,23)(12,17,13,21)(25,30,31,48)(26,35,32,46)(27,37,38,50)(28,43,40,53)(29,45,33,47)(34,49,39,52)(36,51,41,54)(42,55,44,56)

Polycyclic presentation:

2^9|b^2=gh,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, 10125850);