#8199217

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

4392 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)(11,17)(12,22)(13,18)(14,19)(15,16)(20,21)(23,24)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(1,2)(3,7)(4,5)(6,8)(10,16)(13,19)(17,22)(20,23)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(2,5)(6,7)(9,11,15,12)(10,17,16,22)(13,20,19,23)(14,24,18,21)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,40,35,33);(9,12,15,11)(10,17,16,22)(13,23,19,20)(14,24,18,21)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,3)(2,6)(4,8)(5,7)(9,13,15,19)(10,18,16,14)(11,20,12,23)(17,21,22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,18)(10,19,16,13)(11,21,12,24)(17,20,22,23)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)

Polycyclic presentation:

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

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, 8106413);