#8932190

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

4355 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)(4,5)(6,8)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(2,5)(6,7)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(26,31)(27,34)(32,37)(35,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(14,23)(21,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,17,13)(14,20,23,19)(18,24,22,21)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,15,16)(10,12,17,11)(14,21,23,24)(18,20,22,19)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40);(9,14,15,23)(10,18,17,22)(11,19,12,20)(13,21,16,24)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839386);