#9390588

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

4380 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(28,36)(32,40)(34,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,26,30,33)(27,29,35,31)(28,32,36,40)(34,38,37,39);(1,3)(2,4)(5,6)(7,8)(10,17)(11,20)(12,22)(18,24)(25,27,30,35)(26,31,33,29)(28,34,36,37)(32,39,40,38);(1,4,5,8)(2,6,7,3)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,27)(26,29)(28,37)(30,35)(31,33)(32,39)(34,36)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(1,3)(2,4)(5,6)(7,8)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,29)(26,27)(28,38)(30,31)(32,34)(33,35)(36,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297784);