#8612212

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

4545 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)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(26,31)(32,34)(33,38)(39,40);(2,3)(6,7)(10,16)(13,18)(17,20)(22,24)(25,27,30,28)(26,32,31,34)(29,35,37,36)(33,39,38,40);(4,8)(6,7)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,28,30,27)(26,32,31,34)(29,35,37,36)(33,40,38,39);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,29,30,37)(26,33,31,38)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8519408);