#10102694

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

4347 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)(11,16)(12,14)(13,20)(15,19)(17,22)(18,23)(21,24)(25,26,30,31)(27,32,35,38)(28,33,39,36)(29,37,40,34);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,15,12,16)(13,18,22,21)(17,23,20,24)(25,26,30,31)(27,32,35,38)(28,36,39,33)(29,34,40,37);(1,3,5,6)(2,4,7,8)(10,12)(15,16)(17,20)(23,24)(26,31)(27,35)(33,36)(34,37);(1,4)(2,6)(3,7)(5,8)(9,12,14,10)(11,16,19,15)(13,20,22,17)(18,24,21,23)(25,27)(26,32)(28,37)(29,36)(30,35)(31,38)(33,40)(34,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,28)(26,33)(27,34)(29,32)(30,39)(31,36)(35,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,29)(26,34)(27,36)(28,38)(30,40)(31,37)(32,39)(33,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009890);