#8108406

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

4536 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)(6,8)(9,10)(11,12)(13,16)(14,21)(15,17)(18,23)(19,24)(20,22)(26,31)(28,37)(32,38)(35,40);(2,5)(6,8)(9,11,16,15)(10,17,13,12)(14,19,23,22)(18,24,21,20)(27,34)(32,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,13)(14,20)(16,17)(18,22)(19,21)(23,24)(25,26,30,31)(27,32,34,38)(28,33,37,29)(35,39,40,36);(9,13,16,10)(11,17,15,12)(14,21,23,18)(19,24,22,20)(25,27,30,34)(26,32,31,38)(28,35,37,40)(29,36,33,39);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,28)(26,33)(27,35)(29,31)(30,37)(32,39)(34,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,15,16,11)(10,17,13,12)(14,22,23,19)(18,24,21,20)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015602);