#9123924

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9031120);