#10511116

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

4405 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)(12,20)(16,21)(19,22)(23,24)(25,26,32,29)(27,36,38,33)(28,34,39,30)(31,35,37,40);(9,10)(11,18)(12,21)(13,15)(14,17)(16,20)(19,23)(22,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,15,17,18)(12,19,20,22)(16,23,21,24)(25,28,32,39)(26,34,29,30)(27,35,38,40)(31,36,37,33);(3,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,21)(18,24)(25,29,32,26)(27,33,38,36)(28,30,39,34)(31,40,37,35);(1,3)(2,5)(4,7)(6,8)(9,13,14,11)(10,15,17,18)(12,22,20,19)(16,23,21,24)(25,30)(26,28)(27,37)(29,39)(31,38)(32,34)(33,35)(36,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,31,32,37)(26,35,29,40)(27,34,38,30)(28,36,39,33)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=i,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, 10418312);