#9054433

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	#56089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,7)(4,6)(5,8)(9,10,15,12)(11,18,17,14)(13,16,22,20)(19,24,23,21)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,40)(38,49)(39,43)(41,50)(42,47)(44,45)(46,48)(51,53)(52,55)(54,56);(1,2,4,6)(3,5,8,7)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,24,23,21)(26,32)(29,33)(34,47)(36,48)(40,43)(44,49)(50,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,33)(28,39)(29,32)(30,45)(31,35)(34,43)(36,44)(37,42)(38,46)(40,47)(41,51)(48,49)(50,53)(52,54)(55,56);(1,3,4,8)(2,5,6,7)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28,31,42)(26,34,32,47)(27,37,35,39)(29,43,33,40)(30,41,46,54)(36,50,48,56)(38,51,45,52)(44,55,49,53);(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29,31,33)(26,35,32,27)(28,40,42,43)(30,44,46,49)(34,37,47,39)(36,45,48,38)(41,53,54,55)(50,51,56,52);(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30)(26,36)(27,38)(28,41)(29,44)(31,46)(32,48)(33,49)(34,50)(35,45)(37,51)(39,52)(40,53)(42,54)(43,55)(47,56)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gh,e^2=ghi,f^2=h,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, 8961629);