#10247126

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

4498 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154322);