#10529810

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

4506 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)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,26)(27,34)(28,31)(29,32)(30,36)(33,38)(35,39)(37,40);(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(1,2,5,4)(3,6,7,8)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(3,7)(6,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,28,31,26)(27,34,29,32)(30,36,38,33)(35,40,37,39);(1,3,5,7)(2,6,4,8)(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,4,5,2)(3,8,7,6)(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,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, 10437006);