#9053868

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	#47943	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8961064);