#10183080

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

4650 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=ghi,e^2=gi,f^2=g,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, 10090276);