#10482434

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

4448 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(26,27)(29,37)(31,33)(35,38);(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(26,27)(31,33)(32,34)(39,40);(1,2,5,4)(3,6,7,8)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,39,38,40);(3,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,28)(26,31)(27,33)(29,38)(30,36)(32,40)(34,39)(35,37);(1,4,5,2)(3,8,7,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,29,30,37)(26,32,27,34)(28,35,36,38)(31,39,33,40)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,e^2=g,f^2=gh,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, 10389630);