#9173154

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

4588 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,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, 9080350);