#8399318

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,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, 8306514);