#10247256

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154452);