#8393363

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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300559);