#9683967

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

4362 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)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26)(27,33)(28,35)(29,32)(30,31)(34,40)(36,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39);(1,3,5,6)(2,4,7,8)(26,31)(28,33)(32,39)(36,40);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9591163);