#8464346

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

4515 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)(3,4)(5,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(1,3,6,7)(2,4,8,5)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,4,6,5)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,31,40)(26,32,36,37)(28,38,35,29)(30,39,33,34);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28)(26,33)(27,38)(29,40)(30,36)(31,35)(32,34)(37,39);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,37,33,32);(1,5,6,4)(2,7,8,3)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371542);