#9054531

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

4359 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=h,e^2=i,f^2=g,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, 8961727);