#8417410

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
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^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, 8324606);