#8108417

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

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=g,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, 8015613);