#8418634

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=gi,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, 8325830);