#8932921

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,f^2=g,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, 8840117);