#8417326

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

4367 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324522);