#8726708

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gi,e^2=ghi,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, 8633904);