#8398589

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	#39410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=g,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, 8305785);