#9717419

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9624615);