#8636169

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8543365);