#10529227

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gi,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, 10436423);