#9676653

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gh,e^2=g,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, 9583849);