#10529806

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

4416 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10437002);