#10166185

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

4272 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)(3,4)(5,7)(6,8)(9,10,16,14)(11,19,15,17)(12,18,13,22)(20,24,21,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,16,15)(10,17,14,19)(12,20,13,21)(18,23,22,24)(27,36)(29,37)(32,35)(34,40);(1,2,5,7)(3,6,8,4)(9,12,16,13)(10,18,14,22)(11,20,15,21)(17,23,19,24)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(2,7)(4,6)(9,13,16,12)(10,18,14,22)(11,21,15,20)(17,23,19,24)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,4)(2,3)(5,6)(7,8)(9,14,16,10)(11,19,15,17)(12,22,13,18)(20,24,21,23)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,3)(2,6)(4,7)(5,8)(9,15,16,11)(10,17,14,19)(12,21,13,20)(18,23,22,24)(25,30,31,26)(27,35,36,32)(28,38,39,33)(29,34,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gh,e^2=ghi,f^2=gh,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, 10073381);