#8716856

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

4549 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)(11,16)(12,20)(13,17)(14,18)(15,21)(19,23)(22,24)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,2,5,8)(3,4,7,6)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,22)(21,24)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(1,3)(2,6)(4,8)(5,7)(12,21)(15,20)(19,24)(22,23)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(1,4,5,6)(2,7,8,3)(9,12)(10,15)(11,19)(13,23)(14,21)(16,22)(17,24)(18,20)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,22)(20,23)(21,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,4)(2,7)(3,8)(5,6)(9,13,14,16)(10,17,18,11)(12,22,21,23)(15,19,20,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=g,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, 8624052);