#8650393

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,18,13,15)(12,20,22,17)(19,23,21,24)(26,30)(31,37)(32,38)(39,40);(2,3)(6,7)(9,11,14,13)(10,15,16,18)(12,21,22,19)(17,24,20,23)(26,30)(27,33)(32,38)(34,35);(1,2,5,3)(4,6,8,7)(10,16)(15,18)(17,20)(23,24)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(9,13,14,11)(10,15,16,18)(12,19,22,21)(17,24,20,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557589);