#8108315

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

4630 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)(11,18)(12,20)(13,15)(14,16)(17,21)(19,23)(22,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,39)(37,38);(2,5)(6,8)(10,13)(12,21)(16,18)(19,22)(26,31)(32,38)(33,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(25,27,30,37)(26,32,31,38)(28,35,29,36)(33,39,34,40);(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,28,30,29)(26,33,31,34)(27,35,37,36)(32,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,18,14,16)(12,20,21,17)(19,24,22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,14,15,11)(10,18,13,16)(12,22,21,19)(17,24,20,23)(25,29,30,28)(26,34,31,33)(27,36,37,35)(32,40,38,39)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=h,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^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, 8015511);