#8720278

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	#44445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627474);