#8739198

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	#52546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4167 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8646394);