#10506270

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=h,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, 10413466);