#10460703

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

4412 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)(3,5,7,8)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,21,23,20)(25,26,29,32)(27,30,35,33)(28,36,38,31)(34,40,37,39);(9,11,14,19)(10,12,17,15)(13,18,22,23)(16,20,24,21)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,2)(3,5)(4,6)(7,8)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(3,7)(5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,3,4,7)(2,5,6,8)(9,12,14,15)(10,11,17,19)(13,21,22,20)(16,23,24,18)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10367899);