#10412010

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

4559 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)(3,7,8,5)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(26,30)(31,33)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(3,8)(5,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,f^2=g,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, 10319206);