#10045682

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

4537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9952878);