#10420209

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10327405);