#8600821

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

4551 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=g,f^2=h,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, 8508017);