#8341598

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

4459 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248794);