#8406521

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

4637 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)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,31,37)(27,36,39,32)(28,35,30,33)(29,34,40,38);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27)(26,32)(28,38)(29,33)(30,34)(31,39)(35,40)(36,37);(1,3)(2,5)(4,7)(6,8)(10,14)(15,17)(16,20)(23,24)(25,28,31,30)(26,33,37,35)(27,38,39,34)(29,36,40,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,29)(26,34)(27,33)(28,36)(30,32)(31,40)(35,39)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,30,31,28)(26,35,37,33)(27,38,39,34)(29,36,40,32);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,27)(26,36)(28,34)(29,33)(30,38)(31,39)(32,37)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,e^2=g,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, 8313717);