#9059370

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^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, 8966566);