#9720455

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
prime radical	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,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, 9627651);