#9912904

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

4203 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,18)(13,16)(14,22)(15,17)(20,21)(23,24)(25,26,29,32)(27,30,33,37)(28,31,36,40)(34,38,35,39);(1,2,4,6)(3,5,8,7)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(27,33)(30,37)(34,35)(38,39);(1,2)(3,5)(4,6)(7,8)(10,17)(12,22)(16,19)(20,23)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(9,13,15,11)(10,16,17,19)(12,23,22,20)(14,21,18,24)(25,27)(26,30)(28,35)(29,33)(31,39)(32,37)(34,36)(38,40);(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820100);