#8873589

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	#46016	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,7,5)(3,4,8,6)(10,12)(11,17)(16,20)(18,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(2,5)(3,8)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(26,29)(28,37)(31,33)(34,38);(1,3)(2,6)(4,5)(7,8)(11,17)(15,19)(18,23)(22,24)(25,27,30,35)(26,31,29,33)(28,34,37,38)(32,39,36,40);(1,4,7,6)(2,3,5,8)(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,27)(26,31)(28,38)(29,33)(30,35)(32,40)(34,37)(36,39);(1,5,7,2)(3,4,8,6)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,40)(35,38);(1,6,7,4)(2,8,5,3)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780785);