#9382303

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

4435 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=h,e^2=g,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, 9289499);