#10325419

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

4590 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)(9,10,15,13)(11,16,18,12)(14,21,23,17)(19,20,22,24)(25,26)(27,34)(28,31)(29,32)(30,36)(33,38)(35,39)(37,40);(9,11)(10,16)(12,13)(14,22)(15,18)(17,20)(19,23)(21,24)(25,27,31,29)(26,32,28,34)(30,37,38,35)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,13)(11,18)(17,21)(19,22)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(3,7)(6,8)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,22)(23,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,12,18,16)(14,21,23,17)(19,20,22,24)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

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

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, 10232615);