#9057761

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=g,f^2=gi,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, 8964957);