#9863959

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

4335 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9771155);