#10534337

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

4380 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,17)(13,23)(14,19)(16,22)(18,20)(21,24)(25,26,32,29)(27,35,31,33)(28,34,39,30)(36,37,38,40);(2,5)(6,7)(9,11,14,21)(10,15,19,24)(12,20,22,13)(16,23,17,18)(25,27,32,31)(26,33,29,35)(28,36,39,38)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,20)(13,21)(14,22)(15,23)(17,19)(18,24)(25,28,32,39)(26,34,29,30)(27,36,31,38)(33,40,35,37);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,20)(13,21)(14,22)(15,18)(16,19)(23,24)(25,29,32,26)(27,33,31,35)(28,30,39,34)(36,40,38,37);(9,10)(11,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,30)(26,28)(27,37)(29,39)(31,40)(32,34)(33,36)(35,38);(9,13,14,20)(10,18,19,23)(11,12,21,22)(15,16,24,17)(25,31,32,27)(26,33,29,35)(28,38,39,36)(30,37,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=g,f^2=ghi,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, 10441533);