#9794273

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

4295 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)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,32,29)(27,30,33,28)(31,37,39,34)(35,36,40,38);(1,3,6,7)(2,5,8,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,35)(32,33)(34,38)(36,37)(39,40);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28,32,30)(26,33,29,27)(31,36,39,38)(34,40,37,35);(1,4,6,5)(2,7,8,3)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,32,26)(27,28,33,30)(31,37,39,34)(35,36,40,38);(1,5,6,4)(2,7,8,3)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,30,32,28)(26,33,29,27)(31,36,39,38)(34,35,37,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,31,32,39)(26,34,29,37)(27,35,33,40)(28,36,30,38)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 9701469);