#9214258

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

4557 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,13)(14,15)(16,21)(17,24)(19,23)(20,22)(25,26,32,34)(27,28,30,29)(31,39,40,33)(35,37,38,36);(2,6)(3,8)(10,15)(12,17)(16,18)(19,22)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,22)(23,24)(25,28,32,29)(26,27,34,30)(31,36,40,37)(33,35,39,38);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,22)(13,23)(14,21)(15,18)(17,19)(20,24)(25,29,32,28)(26,27,34,30)(31,36,40,37)(33,38,39,35);(1,2)(3,4)(5,6)(7,8)(9,12)(10,13)(11,19)(14,17)(15,24)(16,20)(18,23)(21,22)(25,30,32,27)(26,29,34,28)(31,35,40,38)(33,36,39,37);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,20)(12,15)(14,24)(16,22)(18,19)(21,23)(25,31,32,40)(26,33,34,39)(27,35,30,38)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121454);