#9778480

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

4268 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,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, 9685676);