#9040122

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	#51754	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8947318);