#9390453

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

4242 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,2,4,6)(3,5,8,7)(10,13)(12,21)(15,17)(18,22)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,17)(14,19)(16,24)(18,21)(20,23)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(9,12,14,21)(10,16,13,20)(11,18,19,22)(15,23,17,24)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,24,22,23)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=h,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, 9297649);