#9676619

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

4199 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)(11,18)(12,17)(13,22)(14,15)(16,21)(19,20)(23,24)(26,32)(28,33)(34,39)(36,40);(1,2,5,8)(3,4,7,6)(10,15)(12,22)(16,18)(19,24)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,20,17,23)(25,27,31,29)(26,28,32,33)(30,35,38,37)(34,36,39,40);(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,23)(22,24)(25,28,31,33)(26,27,32,29)(30,36,38,40)(34,35,39,37);(1,3)(2,4)(5,7)(6,8)(9,12,14,22)(10,17,15,13)(11,19,21,24)(16,23,18,20)(25,29,31,27)(26,33,32,28)(30,35,38,37)(34,36,39,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,19,18,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,e^2=g,f^2=h,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, 9583815);