#10218582

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	#47066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(13,16)(15,19)(18,23)(20,21)(22,24)(25,26,31,28)(27,35,38,32)(29,33,41,39)(30,40,43,34)(36,47,49,44)(37,45,51,48)(42,52,53,46)(50,54,56,55);(1,2)(3,7)(4,6)(5,8)(9,11,15,13)(10,16,19,17)(12,22,23,20)(14,21,18,24)(29,41)(33,39)(36,49)(42,53)(44,47)(46,52)(50,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,17,19,16)(12,20,23,22)(14,21,18,24)(25,27,31,38)(26,32,28,35)(29,36,41,49)(30,37,43,51)(33,44,39,47)(34,45,40,48)(42,50,53,56)(46,54,52,55);(2,6)(5,7)(9,12)(10,18)(11,20)(13,22)(14,19)(15,23)(16,21)(17,24)(25,28,31,26)(27,32,38,35)(29,39,41,33)(30,40,43,34)(36,44,49,47)(37,45,51,48)(42,52,53,46)(50,54,56,55);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,16,19,17)(12,20,23,22)(14,21,18,24)(25,29)(26,33)(27,36)(28,39)(30,42)(31,41)(32,44)(34,46)(35,47)(37,50)(38,49)(40,52)(43,53)(45,54)(48,55)(51,56);(1,3)(2,7)(4,8)(5,6)(9,14,15,18)(10,12,19,23)(11,21,13,24)(16,22,17,20)(25,30)(26,34)(27,37)(28,40)(29,42)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(41,53)(44,54)(47,55)(49,56)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,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, 10125778);