#10521390

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gi,e^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, 10428586);