#10481047

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

4243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^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, 10388243);