#10504549

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

4505 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)(10,14)(12,20)(15,17)(18,21)(25,26,32,34)(27,35,30,31)(28,37,29,33)(36,39,38,40);(10,14)(15,17)(16,22)(23,24)(25,27,32,30)(26,31,34,35)(28,36,29,38)(33,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,32,29)(26,33,34,37)(27,36,30,38)(31,39,35,40);(3,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,29,32,28)(26,33,34,37)(27,36,30,38)(31,40,35,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,30,32,27)(26,31,34,35)(28,38,29,36)(33,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,31)(26,30)(27,34)(28,39)(29,40)(32,35)(33,38)(36,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=gi,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, 10411745);