#8726575

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=h,e^2=i,f^2=gi,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, 8633771);