#8424951

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

4388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332147);