#8718385

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625581);