#9039436

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946632);