#8924959

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

4786 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,5)(4,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,27,30,28)(26,31,33,34)(29,36,38,35)(32,40,37,39);(1,3,4,8)(2,5,6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832155);