#9183905

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=gh,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, 9091101);