#10042541

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9949737);