#8726709

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	#59115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,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, 8633905);