#10409926

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

4537 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)(3,7)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,5)(6,8)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,26,30,33)(27,28,31,34)(29,32,38,40)(35,36,37,39);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,27)(26,28)(29,37)(30,31)(32,39)(33,34)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39)

Polycyclic presentation:

2^9|a^2=g,c^2=g,f^2=gh,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, 10317122);