#8599309

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gh,e^2=g,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, 8506505);