#9053925

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

4204 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961121);