#8716882

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

4293 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=gh,f^2=g,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, 8624078);