#8348298

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

4388 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8255494);