#8689232

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26,32,36)(27,34,28,33)(29,31,30,35)(37,40,38,39);(1,2)(3,7)(4,6)(5,8)(10,11)(15,17)(16,18)(23,24)(25,27,32,28)(26,33,36,34)(29,37,30,38)(31,39,35,40);(1,3)(2,5)(4,8)(6,7)(25,28,32,27)(26,34,36,33)(29,38,30,37)(31,40,35,39);(2,6)(5,7)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,29,32,30)(26,35,36,31)(27,37,28,38)(33,40,34,39);(1,3)(2,7)(4,8)(5,6)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,30,32,29)(26,31,36,35)(27,38,28,37)(33,39,34,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,31)(26,30)(27,39)(28,40)(29,36)(32,35)(33,38)(34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596428);