#9634929

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

4246 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=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, 9542125);