#8116812

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

4528 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8024008);