#8691824

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	#49546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,18)(13,22)(14,20)(15,21)(17,23)(19,24)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,21,20)(12,19,23,22)(13,18,24,17)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(9,12)(10,17)(11,22)(13,20)(14,19)(15,23)(16,24)(18,21)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2,4,7)(3,6,8,5)(9,12)(10,18)(11,19)(13,20)(14,22)(15,23)(16,24)(17,21)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,18)(12,20)(14,17)(15,24)(16,23)(21,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,20,21,16)(12,19,23,22)(13,18,24,17)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,36,37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8599020);