#9862948

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770144);