#9183871

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9091067);