#8665283

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

4593 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8572479);