#9061516

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

4462 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)(13,17)(14,15)(19,24)(20,23)(21,22)(25,26,30,32)(27,31,35,38)(28,33,37,40)(29,36,39,34);(1,2,5,6)(3,7,8,4)(10,15)(11,17)(12,20)(22,24)(25,27,30,35)(26,31,32,38)(28,36,37,34)(29,33,39,40);(1,2)(3,7)(4,8)(5,6)(11,17)(12,20)(13,18)(16,23)(26,32)(27,35)(28,37)(29,39);(2,6)(3,8)(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,24,23,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,20)(10,16,15,23)(11,19,17,21)(13,22,18,24)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,40)(34,35);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,23)(20,24)(25,29,30,39)(26,34,32,36)(27,33,35,40)(28,38,37,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8968712);