#9057914

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

4438 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)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,31,29)(27,32,28,35)(30,37,38,33)(34,40,36,39);(2,6)(4,7)(10,15)(12,16)(18,22)(20,23)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(1,2)(3,4)(5,6)(7,8)(11,17)(12,16)(13,21)(18,22)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24);(1,4)(2,3)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965110);