#8417265

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=gi,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, 8324461);