#8651427

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

4557 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8558623);