#10166318

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 10073514);