#8933923

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

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841119);