#10219431

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10126627);