#8471839

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

4719 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379035);